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a “magic” approach to octonionic

Rosenfeld spaces

12 December 2022

Alessio Marrania, Daniele Corradettib, David Chesterc,

Raymond Aschheimc, Klee Irwinc

aInstituto de Física Teorica, Departamento de Física,

Universidad de Murcia, Campus de Espinardo, E-30100, Spain

email: alessio.marrani@um.es

bUniversidade do Algarve, Departamento de Matemática,

Campus de Gambelas, 8005-139 Faro, Portugal

email: a55944@ualg.pt

cQuantum Gravity Research,

Topanga Canyon Rd 101 S., California CA 90290, USA

email: DavidC@QuantumGravityResearch.org;

Raymond@QuantumGravityResearch.org;

Klee@QuantumGravityResearch.org

In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all

real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over

the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfor-

tunately, the procedure carried out by Rosenfeld was not rigorous, since many of the theorems

he had been using do not actually hold true in the case of algebras that are not alternative nor

power-associative. A more rigorous approach to the deﬁnition of all the planes presented more

than thirty years ago by Rosenfeld in terms of their isometry group, can be considered within

the theory of coset manifolds, which we exploit in this work, by making use of all real forms

of Magic Squares of order three and two over Hurwitz normed division algebras and their split

versions. Within our analysis, we ﬁnd 7 pseudo-Riemannian symmetric coset manifolds which

seemingly cannot have any interpretation within Rosenfeld’s framework. We carry out a similar

analysis for Rosenfeld lines, obtaining that there are a number of pseudo-Riemannian symmetric

cosets which do not have any interpretation à la Rosenfeld.

1

arXiv:2212.06426v1 [math.RA] 13 Dec 2022

Contents

1 Introduction 3

2 Real forms of Magic Squares of order 3 and 2 4

2.1 Hurwitz algebras and their split versions . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Trialitiesandderivations................................ 5

2.3 Euclidean and Lorentzian simple rank-3 Jordan algebras . . . . . . . . . . . . . . 6

2.4 Real forms of the Freudenthal-Tits Magic Square . . . . . . . . . . . . . . . . . . 7

2.5 MagicSquareoforder2 ................................ 8

2.6 Octonionic entries of Magic Squares of order 2 and 3 . . . . . . . . . . . . . . . . 8

3 “Magic” formulæ for Rosenfeld planes 9

3.1 Cosetmanifolds..................................... 9

3.2 “Magic”formulæ .................................... 11

4Octonionic Rosenfeld planes 12

4.1 O'R⊗O........................................ 12

4.2 C⊗O.......................................... 13

4.3 H⊗O.......................................... 14

4.4 O⊗O.......................................... 15

5 “Magic” formulæ for Rosenfeld lines 16

6Octonionic Rosenfeld lines 19

6.1 O'R⊗O........................................ 19

6.2 C⊗O.......................................... 19

6.3 H⊗O.......................................... 20

6.4 O⊗O.......................................... 21

7 Conclusions 23

2

1 Introduction

Around the half of the XX century, geometric investigations on the octonionic plane gave rise

to a fruitful mathematical activity, culminating into the formulation of Tits-Freudenthal Magic

Square [Fr65, Ti]. The Magic Square is an array of Lie algebras, whose entries are obtained

from two Hurwitz algebras1and enjoy multiple geometric and algebraic interpretations [BS00,

BS03, SH, Sa, El02, El04], as well as physical applications [GST, DHW, CCM].

The entries m3(A1,A2)of the Tits-Freudenthal Magic Square can be deﬁned equivalently

[BS03] using the Tits [Ti], Barton-Sudbery [BS03] (also cf. [Ev]), and Vinberg [Vin66] construc-

tions2,

m3(A1,A2) =

der(A1)⊕der(J3(A2)) ˙

+ImA1⊗J0

3(A2);

der(A1)⊕der(A2)˙

+sa(3,A1⊗A2);

tri(A1)⊕tri(A2)˙

+ 3(A1⊗A2),

(1)

where A1and A2are normed Hurwitz division algebras R,C,H,Oor the split counterparts

Cs,Hs,Os. Here, J3(A)denotes the Jordan algebra of 3×3Hermitian matrices over Aand

J0

3(A)its subspace of traceless elements. The space of anti-Hermitian traceless n×nmatrices

over A1⊗A2is denoted sa(n, A1⊗A2). Details of the commutators and the isomorphisms

between these Lie algebras can be found e.g. in [BS03]. The choice of division or split A1,A2

yields diﬀerent real forms : for the Tits construction, further possibilities are given by allowing

the Jordan algebra to be Lorentzian, denoted J2,1, as described in [CCM] (see Sec. 2.3 below);

equivalently, one can introduce overall signs in the deﬁnition of the commutators between distinct

components in the Vinberg or Barton-Sudbery constructions, as described in [ABDHN]. For a

complete listing of all possibilities3, see [CCM]. Recently [WDM], Wilson, Dray and Manogue

gave a new construction of the Lie algebra e8in terms of 3×3matrices such that the Lie

bracket has a natural description as matrix commutator; this led to a new interpretation of the

Freudenthal-Tits Magic Square of Lie algebras4, acting on themselves by commutation.

While Tits was more interested in algebraic aspects, Freudenthal interpreted every row of the

Magic Square in terms of a diﬀerent kind of geometry, i.e. elliptic, pro jective, symplectic and

meta-symplectic for the ﬁrst, second, third and fourth row, respectively [LM01]. This means

that, while the algebras taken in account were always the Hurwitz algebras R,C,Hand O, all

Lie groups were arising from considering diﬀerent type of transformations such as isometries,

collineations and homographies of the plane [Fr65]. On the other hand, Rosenfeld conceived

every entry of the 4×4array of the Magic Square as the Lie algebra of the (global) isometry

group of a “generalized” projective plane, later called Rosenfeld plane [Ro98, Ro97]. Thus, while

in Freudenthal’s framework the projective plane was always the same but, depending on the

considered row of the Magic Square, the kind of transformations was changed; in Rosenfeld’s

picture the group of transformations was always the isometry group, and diﬀerent Lie groups

appeared considering diﬀerent planes over tensor product of Hurwitz algebras.

To be more precise, since the exceptional Lie group F4is the isometry group of the octonionic

projective plane OP2, then Rosenfeld regarded E6as the isometry group of the bioctonionic “pro-

1Even though the original construction by Tits was not symmetric in the two entries (cf. the third of (1)),

the square turned out to be symmetric and was, therefore, dubbed as “magic”.

2The ternary algebra approach of [Ka] was generalised by Bars and Günaydin in [BG] to include super Lie

algebras. Generalizations to aﬃne, hyperbolic and further extensions of Lie algebras have been considered in

[Pa].

3Concerning the extension of Freudenthal-Tits Magic Square to various types of “intermediate” algebras, the

extension to the sextonions S(between Hand O) is due to Westbury [Wes], and later developed by Landsberg

and Manivel [LM01, LM06], whereas further extension to the tritonions T(between Cand H) was discussed by

Borsten and one of the present authors in [BM].

4Magic Squares of order 2 of Lie groups have been investigated in [DHK]. Interestingly, the Freudenthal-Tits

Magic Square of Lie groups was observed to be non-symmetric since [Yo85].

3

jective” plane (C⊗O)P2, E7as the isometry group of the quaternoctonionic “projective” plane

(H⊗O)P2, and E8as the isometry group of the octooctonionic “projective” plane (O⊗O)P2.

Despite Rosenfeld’s suggestive interpretation, it was soon realized that the planes identiﬁed by

Rosenfeld did not satisfy projective axioms, and this explains the quotes in the term “projec-

tive”, which would have to be intended only in a vague sense. It was the work of Atsuyama,

followed by Landsberg and Manivel, to give a rigorous description of the geometry arising from

Rosenfeld’s approach [Atsu2, LM01, AB]. Indeed (unlike O'R⊗O)C⊗O,H⊗Oand O⊗O

are not division algebras, thus preventing a direct projective construction; moreover (unlike

C⊗O), Hermitian 3×3matrices over H⊗Oor O⊗Odo not form a simple Jordan algebra, so

the usual identiﬁcation of points (lines) with trace 1 (2) projection operators cannot be made

[Ba]. Nonetheless, they are in fact geometric spaces, generalising projective spaces, known as

“buildings”, on which (the various real forms of) exceptional Lie groups act as isometries. Build-

ings where originally introduced by Tits in order to provide a geometric approach to simple Lie

groups, in particular the exceptional cases, but have since had far reaching implications; see, for

instance, [Ti2, Te] and Refs. therein.

More speciﬁcally, Rosenfeld noticed that, over R, the isometry group of the octonionic pro-

jective plane was the compact form of F4, i.e. F4(−52) , while the other, non-compact real forms,

i.e. the split form F4(4) and F4(−20), were obtained as isometry groups of the split-octonionic

projective plane OsP2and of the octonionic hyperbolic plane OH2[CMCAb, CMCAa], respec-

tively. Thence, he proceeded in relating all real forms of exceptional Lie groups with projective

and hyperbolic planes over tensorial products of Hurwitz algebras [Ro97, Ro98].

Despite being very insightful, to the best of our knowledge Rosenfeld’s approach was never

formulated in a systematic way, and a large part of the very Rosenfeld planes were never really

rigorously deﬁned. In the present investigation, we give a systematic and explicit deﬁnition

of all octonionic5“Rosenfeld planes” as coset manifolds, by exploiting all real forms of Magic

Squares of order 3 over Hurwitz algebras and their split versions, thus all possible real forms

of Freudenthal-Tits Magic Square. In doing so, we will also ﬁnd a total of 10 coset manifolds,

namely 7 “planes” and 3 “lines”, whose deﬁnition is straightforward according to our procedure,

but that apparently do not have any interpretation according to Rosenfeld’s approach.

The plan of the paper is as follows. In Sec. 2 we will resume all the algebraic machinery

used in this paper, i.e. Hurwitz algebras, their triality and derivation symmetries, Euclidean

and Lorentzian cubic Jordan algebras, Magic Squares of order 3 and 2, etc. In Sec. 3 we will

then present the “magic” formulæ which we will use in order to deﬁne the octonionic Rosenfeld

planes in Sec. 4; then, we will do the same for octonionic Rosenﬂed lines in Secs. 5 and 6. Some

ﬁnal remarks and an outlook are given in Sec. 7.

2 Real forms of Magic Squares of order 3 and 2

2.1 Hurwitz algebras and their split versions

Let the octonions Obe the only unital, non-associative, normed division algebra with R8de-

composition of x∈Ogiven by

x=

7

X

k=0

xkik,(2)

where {i0= 1,i1, ..., i7}is a basis of R8and the multiplication rules are mnemonically encoded

in the Fano plane (see left side of Fig. 1), along with i2

k=−1for k= 1, ..., 7. Let the norm

5The quaternionic, complex and real “Rosenfeld planes” and “Rosenfeld lines” can be obtained as proper

sub-manifolds of their octonionic counterparts, so we will here focus only on the latter ones.

4

Figure 1: Multiplication rule of octonions O(left) and of split-octonions Os(right) as real

vector space R8in the basis {i0= 1,i1, ..., i7}. In the case of the octonions i2

0= 1 and i2

k=−1

for k= 1...7, while in the case of split-octonions i2

k= +1, for k= 1,2,3and i2

k=−1for

k6= 1,2,3.

N(x)be deﬁned as

N(x) :=

7

X

k=0

(xkik)2,(3)

and its polarisation as hx, yi:= N(x+y)−N(x)−N(y).Then the real part of xis R(x) :=

hx, 1iand x:= 2 hx, 1i − x. We then have that N(x) = xx, hx, yi=xy +yx,and that

N(xy) = N(x)N(y),(4)

or, in other words, that octonions are a composition algebra with respect to the Norm Ndeﬁned

by (3).

We now deﬁne as the algebra of quaternions Has the subalgebra of Ogenerated by the

elements {i0= 1,i1,i2,i3}and the algebra of complex numbers Cas the subalgebra of Ogenerated

by the elements {i0= 1,i1}.

Thus, the split-octonions Oscan be deﬁned as the only unital, non-associative, normed

algebra with R8decomposition of x∈Osgiven by (2) where, again, {i0= 1,i1, ..., i7}is a basis

of R8, but the multiplication rules are encoded in a diﬀerent variant of the Fano plane, given

by the right side of Fig. 1, along with i2

k= 1 for k= 1, ..., 3and i2

k=−1for k= 4,5,6,7.

Norm and conjugation are deﬁned as in the octonionic case and, as for the division octonions,

we obtain that split octonions are a composition algebra with respect to the Norm Ndeﬁned

by (3). We then deﬁne the algebra of split-quaternions Hsas the subalgebra of Osgenerated by

{i0= 1,i1,i4,i5}and the split-complex (or hypercomplex )numbers Csas the subalgebra of Os

generated by {i0= 1,i4}.

2.2 Trialities and derivations

Let Abe an algebra and End (A)the associative algebra of its linear endomorphisms. Then, the

triality algebra tri (A)is the Lie subalgebra of so (A)⊕so (A)⊕so (A)deﬁned as

tri (A) := n(A, B, C )∈End (A)⊕3:A(xy) = B(x)y+xC (y),∀x, y ∈Ao,(5)

where the Lie bracket are those inherited as a subalgebra. Derivations are a special case of

trialities of the form (A, A, A)and therefore they again form a Lie algebra, deﬁned as

der (A) := {A∈End (A) : A(xy) = A(x)y+xA (y),∀x, y ∈A},(6)

5

R C H O

tri (A)Øu(1) ⊕u(1) so (3) ⊕so (3) ⊕so (3) so (8)

tri (As)Øso (1,1) ⊕so (1,1) sl2(R)⊕sl2(R)⊕sl2(R)so (4,4)

der (A)Ø Ø so (3) g2(−14)

der (As)Ø Ø sl2(R)g2(2)

A(A)Øu1su2Ø

A(As)Øso1,1sl2(R)Ø

Table 1: Triality and derivation algebras of Hurwitz algebras. Moreover we also added the

algebra A(A) := tri (A)so (A),that will be used in Sec. 3.2 in the deﬁnition of the Rosenfeld

planes.

where the bracket is given by the commutator. In the case of A=C,H,Owe intended so (A)

as so (2),so (4) and so (8) respectively; while, for their split companions, we intended so (As)as

so (1,1),so (2,2) and so (4,4). Triality and derivation Lie algebras over Hurwitz algebras are

summarized in Table 1.

2.3 Euclidean and Lorentzian simple rank-3 Jordan algebras

Let Abe a composition algebra and Hn(A)be the set of Hermitian n×nmatrices with elements

in Asuch that

X†:= Xt=X, (7)

where the conjugation is the one pertaining to Aitself. When6n= 2 or 3, we deﬁne the simple,

rank-nEuclidean Jordan algebra Jn(A)as the commutative algebra over Hn(A)with the Jordan

product ◦deﬁned by the anticommutator

X◦Y:= 1

2(XY +Y X) =: {X, Y },∀X, Y ∈Hn(A),(8)

where juxtaposition denotes the standard ’rows-by-columns’ matrix product. Moreover, we

deﬁne J0

n(A)as the subspace of the Jordan algebra orthogonal to the identity I(namely, the

subspace of traceless matrices in Jn(A)), endowed with the product inherited7from Jn(A)(see

e.g. [BS03, p.8]). In particular, for n= 3 the simple, rank-3Euclidean Jordan algebra J3(A)

has elements

X:=

a x1x2

x1b x3

x2x3c

∈J3(A),(9)

where x1, x2, x3∈Aand a, b, c ∈R.

The Hermiticity condition (7) can be generalized by inserting a pseudo-Euclidean metric :

ηX†η=X, (10)

η:= diag

−1, ..., −1

| {z }

ptimes

,1, ..., 1

| {z }

n−ptimes

.(11)

Correspondingly, one deﬁnes the simple, rank-npseudo-Euclidean Jordan algebra Jn−p,p (A)'

Jp,n−p(A). In particular, by setting n= 3, the simple, rank-3Lorentzian Jordan algebra

6Excluding A=Oand Os, one can actually consider any n∈N.

7Note that J0

n(A)is not closed under the restriction of the Jordan product ◦of Jn(A)to J0

n(A)itself. However,

one can deform ◦into the Michel-Radicati product ∗, deﬁned as X∗Y:= X◦Y−Tr(X◦Y)

nI∀X, Y ∈J0

n(A),

under which J0

n(A)is closed [MR70, MR73].

6

J2,1(A)'J1,2(A), whose elements read as

X:=

a x1x2

−x1b x3

−x2x3c

∈J1,2(A),(12)

where x1, x2, x3∈Aand a, b, c ∈R.

2.4 Real forms of the Freudenthal-Tits Magic Square

As anticipated in (1), given A1and A2two composition algebras, the corresponding entry of

Tits-Freudenthal Magic Square m3(A1,A2)is a Lie algebra that can be realized in at least three

diﬀerent, but equivalent ways, on which we will now brieﬂy comment (without the explicit

deﬁnition of the Lie brackets, that can be found e.g. in [BS03]).

1. Tits construction [Ti] is given by the ﬁrst line of the r.h.s. of (1) :

m3(A1,A2) := der(A1)⊕der(J3(A2)) ˙

+ImA1⊗J0

3(A2),(13)

thus involving the derivation Lie algebra of A1and the derivation Lie algebra of the simple,

rank-3Euclidean Jordan algebra J3(A2)over A2. Tits construction, despite being not

manifestly symmetric under the exchange A1↔A2, is the most general of the three

constructions presented here, since it holds for any alternative algebras A1and A2, as

long as it is possible to deﬁne a Jordan algebra over A2itself. As mentioned above, Tits

construction can be generalized to J2,1(A2), thus yielding8m1,2(A1,A2).

2. A more symmetric approach was pursued by Vinberg [Vin66], who obtained the formula

given by the second line of the r.h.s. of (1),

m3(A1,A2) = der(A1)⊕der(A2)˙

+sa(3,A1⊗A2).(14)

This formula is manifestly symmetric under the exchange A1↔A2, but it only holds if

A1and A2are both composition algebras; it involves only the derivation Lie algebras of A

and B, as well as the 3×3antisymmetric matrices over A1⊗A2. As mentioned above, a

suitable deformation of (14) is possible, in order to give rise to m1,2(A1,A2).

3. Another symmetric formula under the exchange A1↔A2was obtained by Barton and

Sudbery [BS03] (also cf. [Ev]), and it is given by the third line of the r.h.s. of (1),

m3(A1,A2) = tri(A1)⊕tri(A2)˙

+ 3(A1⊗A2),(15)

involving the triality algebras tri (A1)and tri (A2), together with three copies of the tensor

product A1⊗A2. Again, as mentioned above, a suitable deformation of (15) is possible, in

order to give rise to m1,2(A1,A2).

Other, diﬀerent versions of the construction of Freudenthal-Tits Magic Square have been devel-

oped by Santander and Herranz [SH, Sa], Atsuyama [Ats, Atsu2] and Elduque [El04, El02, El18],

all involving composition algebras (even though Elduque’s construction involves ﬂexible compo-

sition algebras instead of alternative composition algebras [El18]).

8Since J2,1(A2)'J1,2(A2), it holds that m2,1(A1,A2)'m1,2(A1,A2).

7

2.5 Magic Square of order 2

Inspired by the works of Freudenthal and Tits, Barton and Sudbery [BS03] also considered a

diﬀerent Magic Square, based on 2×2matrices, and deﬁned by the following formula :

m2(A1,A2) := so (A0

1)⊕der (J2(A2)) ⊕(A0

1⊗J0

2(A2)) ,(16)

which also enjoys a “Vinberg-like” equivalent version as

m2(A1,A2) = so (A0

1)⊕so (A0

2)⊕sa2(A1⊗A2),(17)

that was more recentely used at Lie group level in [DHK]. Formula (16) can be generalized to

involve simple, rank-2 Lorentzian Jordan algebras J1,1(A2)(see Sec. 2.3), thus obtaining the

Lorentzian version of the Magic Square of order 2,

m1,1(A1,A2) := so (A0

1)⊕der (J1,1(A2)) ⊕A0

1⊗J0

1,1(A2),(18)

which we will explicitly evaluate in the treatment below (for the ﬁrst time in literature, to the

best of our knowledge).

2.6 Octonionic entries of Magic Squares of order 2 and 3

In order to rigorously deﬁne all possible octonionic “Rosenfeld planes” and “Rosenfeld lines”

over R(i.e. all possible real forms thereof ), we need to isolate all octonionic and split-octonionic

entries of all real forms of the Freudenthal-Tits Magic Square (order 3) [CCM, BM, ABDHN,

BS03] and of the Magic Square of order 2 [BS03], i.e. we need to consider the entries mα(A1,A2)

in which at least one of A1and A2is Oor Os, for all possible real forms, namely for α= 3

(Euclidean order 3), 1,2(Lorentzian order 3), 2(Euclidean order 2) and 1,1(Lorentzian order

2).

The octonionic entries of the Euclidean Magic Squares of order 3 m3(i.e., of all Euclidean

real forms of Freudenthal-Tits Magic Square) are

Am3(A,O)m3(As,O)m3(Os,A)m3(As,Os)

Rf4(−52) f4(−52) f4(4) f4(4)

Ce6(−78) e6(−26) e6(2) e6(6)

He7(−133) e7(−25) e7(−5) e7(7)

Oe8(−248) e8(−24) e8(−24) e8(8)

The entries of the table above comprise all real forms of e8and e7, and most of the real

forms of e6and f4. The missing real forms f4(−20) and e6(−14) can be recovered as the octonionic

entries of the Lorentzian Magic Squares of order 3 m2,1(i.e., of all Lorentzian real forms of

Freudenthal-Tits Magic Square), given by

Am1,2(A,O)m1,2(As,O)m1,2(Os,A)m1,2(As,Os)

Rf4(−20) f4(−20) f4(4) f4(4)

Ce6(−14) e6(−26) e6(2) e6(6)

He7(−5) e7(−25) e7(−5) e7(7)

Oe8(8) e8(−24) e8(−24) e8(8)

On the other hand, the octonionic entries of the Euclidean Magic Squares of order 2 m2are9

9The explicit form of m2(As,Bs)is, as far as we know, not present in the current literature. We will present

a detailed treatment elsewhere, and here we conﬁne ourselves to report its octonionic column only.

8

Am2(A,O)m2(As,O)m2(Os,A)m2(As,Os)

Rso (9) so (9) so (5,4) so (5,4)

Cso (10) so (9,1) so (6,4) so (5,5)

Hso (12) so (10,2) so (8,4) so (6,6)

Oso (16) so (12,4) so (12,4) so (8,8)

Finally, the octonionic entries of the Lorentzian Magic Squares of order 2 m1,1are given by10

Am1,1(A,O)m1,1(As,O)m1,1(Os,A)m1,1(As,Os)

Rso (8,1) so (8,1) so (5,4) so (5,4)

Cso (8,2) so (9,1) so (6,4) so (5,5)

Hso (8,4) so (10,2) so (8,4) so (6,6)

Oso (8,8) so (12,4) so (12,4) so (8,8)

3 “Magic” formulæ for Rosenfeld planes

In his study of the geometry of Lie groups [Ro97], Rosenfeld deﬁned its “projective” planes

(A⊗B)P2as the completion of some aﬃne planes obtained as non-associative modules over the

tensor algebra A⊗B. He then argued the form of the matrices composing the linear transfor-

mations associated with the Lie algebra of the collineations that preserved the polarity, i.e. the

isometries of the plane. This approach allowed him to relate all real forms of exceptional Lie

groups [Ro93, Ro98] with isometries of suitable “projective” hyperbolic spaces. As mentioned

above, unfortunately the procedure carried out by Rosenfeld was not rigorous, since many of

theorems used in [Ro97] do not extend to the case of algebras that are not alternative nor of

composition, as it is in the case of many algebras listed in Table 2.

We present here a general and rigorous way to deﬁne the spaces considered by Rosenfeld, in

terms of their isometry and isotropy groups, namely using the theory of coset manifolds.

3.1 Coset manifolds

Since our “magic” formulæ deﬁne all Rosenfeld planes as coset manifolds, it is worth brieﬂy

reviewing them, and their relation to homogenous spaces. An homogeneous space is a manifold

on which a Lie group acts transitively, i.e. a manifold on which is deﬁned an action ρgfrom

G×Min Msuch that ρe(m) = mfor ethe identity in Gand m∈Mand for which, given

any m, n ∈Mit exists a a not necessarely unique g∈Gsuch that ρg(m) = n. Within this

framework, the isotropy group Isotm(G)is the set formed by the elements of Gthat ﬁx the

point m∈Munder the action of G, i.e.

Isotm(G) = {g∈G:ρg(m) = m}.(19)

Since, by deﬁnition of group action, we have that ρe(m) = mand ρgh (m) = ρg(ρh(m)), then

K=Isotm(G)is a closed subgroup of Gand, moreover the natural map from the quotient space

G/K in Mgiven by gK 7→ gm is a diﬀeomorphism (see [Ar] and Refs. therein).

Given a Lie group Gand a closed subgroup K < G, then the coset space G/K ={gK :g∈G}

is endowed with a natural manifold structure inherited by Gand is, therefore, called a coset

manifold. Notice that the action of Gon the coset manifold G/K, given by the translation τg

deﬁned as

τg(m) = gm, (20)

10The Lorentzian magic square of order two yield to three algebras, i.e. so (8,1),so (8,2) and so (8,4) which

are not covered in other magic squares. The explicit forms of m1,1(A,B),m1,1(As,B)and m1,1(As,Bs)are, as

far as we know, not present in the current literature. We will present a detailed treatment elsewhere, and here

we conﬁne ourselves to report their octonionic rows and columns only.

9

Algebra Comm. Ass. Alter. Flex. Pow. Ass.

C⊗CYes Yes Yes Yes Yes

C⊗HNo Yes Yes Yes Yes

H⊗HNo Yes Yes Yes Yes

C⊗ONo No Yes Yes Yes

H⊗ONo No No No No

O⊗ONo No No No No

Table 2: Properties of the algebra A⊗Bwhere A,Bare Hurwitz algebras. As for the property

an algebra Ais said to be commutative if xy =yx for every x, y ∈X; it is deﬁned as associative

if satisﬁes x(yz)=(xy)z;alternative if x(yx)=(xy)x;ﬂexible if x(yy)=(xy)yand, ﬁnally,

power-associative if x(xx)=(xx)x.

for every g∈Gand m∈G/K, is transitive, i.e. for every m, n ∈G/K it exists a (not necessarily

unique) g∈Gsuch that n=gm, and thus the coset manifold G/K is an homogenous space. On

the other hand, since multiple groups can act transitively on the same manifold with diﬀerent

isotropy groups, then an homogeneous space can be realised in multiple way as a coset manifold.

Moreover, a close look to the deﬁnitions shows that the isotropy group Isotm(G/K)is exactly

the closed subgroup K.11 In general, the holonomy subgroup and the isotropy subgroup have

the same identity-connected component; so, if one assumes that G/K is simply-connected, they

are equal (see e.g. [Be, He] and Refs. therein).

Moreover, let Gbe a Lie group and Ka closed and connected subgroup of G, denoting with

gand ktheir respective Lie algebras, then the coset manifold G/K is reductive if there exists a

subspace msuch that g=k⊕mand

([k,k]⊂k,

[k,m]⊂m,(21)

while if in addition to (21) we also have

[m,m]⊂k,(22)

then the space is symmetric. All Rosenfeld planes are symmetric coset manifolds.

It is also worth noting that for any coset manifold G/K the structure constants of the Lie

algebra gof the Lie group Gdeﬁne completely the structure constants of the manifold, thus the

invariant metrics, and all the metric-dependent tensors, such as the curvature tensor, the Ricci

tensor, etc. Indeed, let {E1, ..., En}be a basis for gin such a way that {E1, ..., Em}are a basis

for k, which we will also call {K1, ..., Km}for readibility reasons, and {Em+1, ..., En}a basis

for mthat we will also denote as {Mm+1, ..., Mn}. Then consider the structure constants of the

algebra g, i.e.

[Ej, Ek] =

n

X

i=1

Ci

jk Ei,(23)

for every j, k ∈ {1, ..., n}. Conditions for reductivity in (21) are then translated in

[Kj, Kk] =

m

P

i=1

Ci

jk Kifor j, k ∈ {1, ..., m},

[Kj, Mk] =

n

P

i=m+1

Ci

jk Mifor j∈ {1, ..., m}, k ∈ {m+ 1, ..., n},

(24)

11In particular, the origin of G/K is, by deﬁnition, the point at which the K-invariance is immediately manifest.

10

while, on the other hand, from

[Mj, Mk] =

m

X

i=1

Ci

jk Ki+

n

X

i=m+1

Ci

jk Mi,(25)

we deduce that if Ci

jk = 0 for all i, j, k ∈ {m+ 1, ..., n}, we have also a symmetric space. From

the structure constants Ci

jk , all geometrical invariants of the coset manifold can be obtained

(see [FF17] for all technical details) such as the Riemann tensor over the coset manifold G/K

which is given by

Ra

bcd =

n

X

e=m+1

1

λ21

8Ca

edCe

bc −1

8λCa

beCe

cd −Ca

ecCe

bd−1

2λ2

m

X

i=1

Ca

biCi

cd,(26)

and that, in case of symmetric spaces, i.e. when Ci

jk = 0 for i, j, k ∈ {m+ 1, ..., n}, reduces

drastically to

Ra

bcd =−1

2λ2

m

X

i=1

Ca

biCi

cd,(27)

for every a, b, c, d ∈ {m+ 1, ..., n}.

In fact, what is relevant for our purposes is that the Lie group G, which acts as isometry

group, and its closed subgroup K, which acts as isotropy group, deﬁne completely all the metric

properties and geometric invariants of the Rosenfeld planes. On the other hand, instead of

working on Lie groups Gand K, i.e. with the isometry group and the isotropy group respectively,

in the following sections we will work with their respective Lie algebras gand k, in order to recover

the appropriate Gand Kand thus deﬁne the Rosenfeld plane.

3.2 “Magic” formulæ

As seen in the previous section, the geometry of a coset manifold is fully determined by the

(global) isotropy Lie group Gand by its (local) subgroup given by the isotropy Lie group K.

We now characterize the Rosenfeld “projective” planes by specifying the real form of the Lie

groups that will act as isometry and isotropy groups, while the Lie brackets - and therefore the

metrical properties of the space - are those arising from Tits construction. It is here worth noting

that the classiﬁcation of the real forms of simple Lie groups makes use of the character χ, deﬁned

as the diﬀerence of the cardinality of non-compact and compact generators, i.e. χ:= #nc −#c;

consequently, Rosenfeld planes will have a corresponding χ.

Let Aand Bbe two Hurwitz algebras, in their division or split versions. Moreover, let the

Lie group A(A)be such that its Lie algebra is Lie(A(A)) ≡A(A) := tri (A)so (A), cfr. Table 1.

Moreover, let Mα(A,B)be the Lie group with Lie algebra given by the (A,B)-entry of the Magic

Square mα(A,B), namely12 Lie(Mα(A,B)) = mα(A,B). In order to characterize “projective”

Rosenfeld planes over tensor products of Hurwitz algebras in terms of coset manifolds with

isometry (resp. isotropy) Lie groups whose Lie algebras are entries of real forms of the Magic

Square of order 3 (resp. 2), we now introduce the following three diﬀerent classes of locally

symmetric, (pseudo-)Riemannian coset manifolds, that we name as Rosenfeld planes :

1. The projective Rosenfeld plane

(A⊗B)P2'M3(A,B)

M2(A,B)⊗ A (A)⊗ A (B).(28)

12For the pseudo-orthogonal Lie algebras, we will generally consider the spin covering of the corresponding Lie

group.

11

2. The hyperbolic Rosenfeld plane

(A⊗B)H2'M1,2(A,B)

M2(A,B)⊗ A (A)⊗ A (B).(29)

3. The pseudo-Rosenfeld plane

(A⊗B)e

H2'M1,2(A,B)

M1,1(A,B)⊗ A (A)⊗ A (B).(30)

Eqs. (28), (29) and (30) are named “magic” formulæ, since they characterize the Rosenfeld

planes as homogeneous (symmetric) manifolds, with isometry (resp. isotropy) groups whose

Lie algebras are given by the entries of some real forms of the Magic Square of order 3 (resp.

2), with further isotropy factors given by the Lie groups Aassociated to the algebras Aand B

deﬁning the tensor product associated to the class of Rosenfeld plane under consideration. As

previously noticed, the term “projective” and “hyperbolic” are here to be intended in a vague

sense since none of the octonionic Rosenfeld planes with dimension greater than 16 satisfy axioms

of projective or hyperbolic geometry. Nevertheless, those adjectives are not arbitrary since the

notion of such projective planes can be made precise as in [Atsu2, AB] and it then agrees with

the above deﬁnitions.

4Octonionic Rosenfeld planes

We now consider the “magic” formulæ (28)-(30) in the cases13 in which Aand/or Bis Oor

Os: this will allow us to rigorously introduce the octonionic Rosenfeld planes, which all share

the fact that their isometry group is a real form of an exceptional Lie group of F- or E- type;

however, it is here worth anticipating that a few real forms of Rosenfeld planes with exceptional

isometry groups cannot be characterized in this way.

4.1 O'R⊗O

The simplest case concerns the tensor product R⊗O, which is nothing but O: in fact, this

was the original observation by Rosenfeld that started it all, yielding to the usual octonionic

projective, split-octonionic and hyperbolic plane, i.e. OP2,OH2and OsP2, respectively. Within

the framework introduced above, the starting point is given by the Cayley-Moufang plane over

C, namely by the octonionic projective plane over C, i.e. by the locally symmetric coset manifold

having as isometry group the complex form of F4, and as isotropy group Spin(9,C), i.e.

OP2

C'FC

4

Spin (9,C).(31)

In this coset space formulation, the tangent space of OP2

Ccan be identiﬁed with the 16C-

dimensional spinor representation space of the isotropy group Spin(9,C).

Then, by specifying the formulæ (28), (29) and (30) for A=Rand B=Oor Os, we obtain

all real forms14 of (31) as Rosenfeld planes over R⊗O'Oor over R⊗Os'Os; they are

summarized by the following15 Table [CMCAb] :

13As mentioned above, this restriction does not imply any loss of generality, as far as the other, non-octonionic

Rosenfeld planes can be obtained as suitable sub-manifolds of the octonionic Rosenfeld planes.

14By real forms of OP2

C, we here mean the cosets with isometry groups given by all real (compact and non-

compact) forms of F4, and with isotropy group given by all (compact and non-compact) real forms of Spin(9)

which are subgroups of the corresponding real form of F4.

15Recall that A(R) = A(O) = A(Os) = ∅.

12

Plane Isometry Isotropy #nc #cχ

OP2F4(−52) Spin (9) 0 16 −16

OH2F4(−20) Spin (9) 16 0 16

Oe

H2F4(−20) Spin (8,1) 8 8 0

OsP2F4(4) Spin (5,4) 8 8 0

along with

OsP2'OsH2'Ose

H2.(32)

4.2 C⊗O

In the bioctonionic case, i.e. for C⊗O, the starting point is given by the bioctonionic projective

plane over C, i.e. by the locally symmetric coset manifold having as isometry group the complex

form of E6, and as isotropy group Spin(10,C)⊗U1, i.e.

(C⊗O)P2

C'EC

6

Spin (10,C)⊗(U1)C

,(33)

which is a Kähler manifold, and has been recently treated in [CMCAa]. In this case, the

tangent space of the coset manifold (33) is the 16C,+⊕16C,−representation of the isotropy

group Spin (10,C)⊗U1.

Then, by specifying the formulæ (28), (29) and (30) for A=Cor Csand B=Oor Os, we ob-

tain all real forms16 of (33) which can be expressed as Rosenfeld planes over C(or Cs)⊗O(or Os);

they are summarized by the following17 Table :

Plane Isometry Isotropy #nc #cχ

(C⊗O)P2E6(−78) Spin (10) ⊗U10 32 −32

(C⊗O)H2E6(−14) Spin (10) ⊗U132 0 32

(C⊗O)e

H2E6(−14) Spin (8,2) ⊗U116 16 0

(C⊗Os)P2E6(2) Spin (6,4) ⊗U116 16 0

(Cs⊗O)P2E6(−26) Spin (9,1) ⊗SO (1,1) 16 16 0

(Cs⊗Os)P2E6(6) Spin (5,5) ⊗SO (1,1) 16 16 0

along with

(C⊗Os)P2'(C⊗Os)H2'(C⊗Os)e

H2,(34)

(Cs⊗O)P2'(Cs⊗O)H2'(Cs⊗O)e

H2,(35)

(Cs⊗Os)P2'(Cs⊗Os)H2'(Cs⊗Os)e

H2,(36)

expressing the fact that projective, hyperbolic and pseudo Rosenfeld planes involving Csand/or

Osare all isomorphic.

The ﬁrst four manifolds of the above Table, having a U1factor in the isotropy group, are

Kähler manifolds, whereas the last two, having a SO(1,1) factor in the isotropy group, are

pseudo-Kähler manifolds.

16By real forms of (C⊗O)P2

C, we here mean the cosets with isometry groups given by all real (compact

and non-compact) forms of E6, and with isotropy group given by all (compact and non-compact) real forms of

Spin(10,C)⊗(U1)Cwhich are subgroups of the corresponding real form of E6.

17Recall that A(C) = u1, and A(Cs) = so1,1.

13

Finally, and more importantly, there are two real forms of (33), namely the locally symmetric,

pseudo-Riemannian coset Kähler manifolds

X32,I := E6(2)

SO∗(10) ⊗U1

,#nc = 20,#c= 12 ⇒χ= 8; (37)

X32,II := E6(−14)

SO∗(10) ⊗U1

,#nc = 12,#c= 20 ⇒χ=−8,(38)

whose isotropy Lie group has the corresponding Lie algebra which is not an entry of any real

form of the Magic Square of order 2.

In other words, since the Lie algebra so∗(10) does not occur in any real form of the Magic

Square of order 2 (see Secs. 2.5 and 2.6), the symmetric Kähler manifolds (37) and (38) cannot

be characterized as Rosenfeld planes over C(or Cs)⊗O(or Os).

4.3 H⊗O

In the quaternoctonionic case, i.e. for H⊗O, the starting point is given by the quaternoctonionic

“projective” plane over C, i.e. by the locally symmetric coset manifold having as isometry group

the complex form of E7, and as isotropy group Spin(12,C)⊗SL (2,C), i.e.

(H⊗O)P2

C'EC

7

Spin (12,C)⊗SL (2,C),(39)

which is a quaternionic Kähler manifold, and whose tangent space is given by the 32(0),2C

representation18 of the isotropy group Spin(12,C)⊗SL (2,C).

Then, by specifying the formulæ (28), (29) and (30) for A=Hor Hsand B=Oor Os, we ob-

tain all real forms19 of (39) which can be expressed as Rosenfeld planes over H(or Hs)⊗O(or Os);

they are summarized by the following20 Table :

Plane Isometry Isotropy #nc #cχ

(H⊗O)P2E7(−133) Spin (12) ⊗SU (2) 0 64 −64

(H⊗O)H2E7(−5) Spin (12) ⊗SU (2) 64 0 64

(H⊗O)e

H2E7(−5) Spin (8,4) ⊗SU (2) 32 32 0

(Hs⊗O)P2E7(−25) Spin (10,2) ⊗SL (2,R) 32 32 0

(Hs⊗Os)P2E7(7) Spin (6,6) ⊗SL (2,R) 32 32 0

along with

(H⊗O)e

H2'(H⊗Os)P2'(H⊗Os)H2'(H⊗Os)e

H2,(40)

(Hs⊗O)P2∼

=(Hs⊗O)H2∼

=(Hs⊗O)e

H2,(41)

(Hs⊗Os)P2∼

=(Hs⊗Os)H2∼

=(Hs⊗Os)e

H2.(42)

The planes that do not involve split algebras, namely the ones having a SU(2) factor in the

isotropy group, are quaternionic Kähler manifolds, whereas all the ones involving split algebras,

namely the ones having a SL(2,R)factor in the istropy group, are para-quaternionic Kähler

manifolds.

18For the possible priming of the semispinor 32 of Spin(12), see e.g. [Min].

19By real forms of (H⊗O)P2

C, we here mean the cosets with isometry groups given by all real (compact

and non-compact) forms of E7, and with isotropy group given by all (compact and non-compact) real forms of

Spin(12,C)⊗SL2(C)which are subgroups of the corresponding real form of E7.

20Recall that A(H) = su2, and A(Hs) = sl2(R).

14

Again, there are three real forms of (39), namely the locally symmetric, pseudo-Riemannian

(para-)quaternionic coset manifolds

X64,I := E7(7)

SO∗(12) ⊗SU (2),#nc = 40,#c= 24 ⇒χ= 16; (43)

X64,II := E7(−5)

SO∗(12) ⊗SL (2,R),#nc = 32,#c= 32 ⇒χ= 0; (44)

X64,II I := E7(−25)

SO∗(12) ⊗SU (2),#nc = 24,#c= 40 ⇒χ=−16,(45)

whose isotropy Lie group -up to the factor SU(2) or SL(2,R)- has the corresponding Lie algebra

which is not an entry of any real form of the Magic Square of order 2.

In other words, since the Lie algebra so∗(12) does not occur in any real form of the Magic

Square of order 2 (see Secs. 2.5 and 2.6), the symmetric (para-)quaternionic Kähler manifolds

(43)-(45) cannot be characterized as Rosenfeld planes over H(or Hs)⊗O(or Os). It should however

be noticed that the isotropy group of (43)-(45) admits an interpretation in terms of real forms

of the Magic Square of order 3, namely

X64,I 'M3(Hs,Os)

M3(Hs,H)⊗ A(H),(46)

X64,II 'M3(H,Os)

M3(H,Hs)⊗ A(Hs),(47)

X64,II I 'M3(Hs,O)

M3(Hs,H)⊗ A(H),(48)

but still the rationale (if any) of such formulæ is missing, and it is surely not the one underlying

the “magic” formulæ (28)-(30).

4.4 O⊗O

In the octooctonionic case, i.e. for O⊗O, the starting point is given by the octoooctonionic

“projective” plane over C, i.e. by the locally symmetric coset manifold having as isometry group

the complex form of E8, and as isotropy group Spin(16,C), i.e.

(O⊗O)P2

C'EC

8

Spin (16,C),(49)

whose tangent space is given by the 128(0)Crepresentation21 of Spin (16,C).

Then, by specifying the formulæ (28), (29) and (30) for A=Oor Osand B=Oor Os, we ob-

tain all real forms22 of (49) which can be expressed as Rosenfeld planes over O(or Os)⊗O(or Os);

they are summarized by the following Table :

Plane Isometry Isotropy #nc #cχ

(O⊗O)P2E8(−248) Spin (16) 0 128 −128

(O⊗O)H2E8(8) Spin (16) 128 0 128

(O⊗O)e

H2E8(8) Spin (8,8) 64 64 0

(Os⊗O)P2E8(−24) Spin (12,4) 64 64 0

21Again, for the possible priming of the semispinor 128 of Spin(16), see e.g. [Min].

22By real forms of (O⊗O)P2

C, we here mean the cosets with isometry groups given by all real (compact

and non-compact) forms of E8, and with isotropy group given by all (compact and non-compact) real forms of

Spin(16,C)which are subgroups of the corresponding real form of E8.

15

along with

(O⊗O)e

H2'(Os⊗Os)P2'(Os⊗Os)H2'(Os⊗Os)e

H2,(50)

(O⊗Os)P2'(O⊗Os)H2'(O⊗Os)e

H2.(51)

Again, there are two real forms of (49), namely the locally symmetric, pseudo-Riemannian

coset manifolds

X128,I := E8(8)

SO∗(16),#nc = 72,#c= 56 ⇒χ= 16; (52)

X128,II := E8(−24)

SO∗(16),#nc = 56,#c= 72 ⇒χ=−16,(53)

whose isotropy Lie group has the corresponding Lie algebra which is not an entry of any real

form of the Magic Square of order 2.

In other words, since the Lie algebra so∗(16) does not occur in any real form of the Magic

Square of order 2 (see Secs. 2.5 and 2.6), the symmetric manifolds (52)-(53) cannot seemingly

be characterized as Rosenfeld planes over O(or Os)⊗O(or Os). It should however be noticed that

the isotropy group SO∗(16) of (52)-(53) admits an interpretation in terms of the Magic Square

of order 4, namely

X128,I 'M3(Os,Os)

M4(H,Hs);(54)

X128,II 'M3(O,Os)

M4(H,Hs),(55)

but still the rationale (if any) of such formulæ is missing, and it is surely not the one underlying

the “magic” formulæ (28)-(30).

5 “Magic” formulæ for Rosenfeld lines

Let Spin(A)the spin covering Lie group whose Lie algebra is Lie(Spin (A)) = so (A), namely

the Lie algebra which preserves the norm of the Hurwitz algebra A. In order to characterize

“projective” Rosenfeld lines over tensor products of Hurwitz algebras in terms of coset manifolds

with isometry and isotropy Lie groups whose Lie algebras respectively are entries of real forms of

the Magic Square of order 2 and norm-preserving Lie algebras of the involved Hurwitz algebras,

we now introduce a variation of (28) and (29), giving rise to the following two diﬀerent classes

of locally symmetric, (pseudo-)Riemannian coset manifolds, that we name as Rosenfeld lines :

1. The “ projective” Rosenfeld line

(A⊗B)P1'M2(A,B)

Spin (A)⊗Spin (B).(56)

2. The hyperbolic Rosenfeld line

(A⊗B)H2'M1,1(A,B)

Spin (A)⊗Spin (B).(57)

Eqs. (56) and (57) are named “magic” formulæ, since they characterize the Rosenfeld lines as

homogeneous (symmetric) manifolds, with isometry and isotropy Lie groups respectively given

by the entries of some real forms of the Magic Square of order 2 and by norm-preserving Lie

algebras of the involved Hurwitz algebras.

16

Figure 2: Cartan classiﬁcation, Satake diagram, character χof exceptional Lie groups F4and

E6and related octonionic projective or hyperbolic Rosenfeld plane of which they are isometry

group.

17

Figure 3: Cartan classiﬁcation, Satake diagram, character χof exceptional Lie groups E7and

E8and related octonionic projective or hyperbolic Rosenfeld plane of which they are isometry

group.

18

6Octonionic Rosenfeld lines

We now consider the “magic” formulæ (56)-(57) in the cases23 in which Aand/or Bis Oor Os:

this will allow us to rigorously introduce the octonionic Rosenfeld lines; however, again, it is here

worth anticipating that a few real forms of octonionic Rosenfeld lines cannot be characterized

in this way.

6.1 O'R⊗O

Within the framework introduced above, the starting point is given by the octonionic projective

line over C, which is nothing but the 8-sphere S8

Cover C, i.e. the locally symmetric coset

manifold having as isometry group Spin(9,C)and as isotropy group Spin(9,C):

OP1

C'Spin (9,C)

Spin (8,C).(58)

In this coset space formulation, the tangent space of OP1

Ccan be identiﬁed with the 8(v,s,c),C

representation of Spin (8,C)where d4-triality [Por] allows to equivalently choose v,s, or c.

Then, by specifying the formulæ (56) and (57) for A=Rand B=Oor Os, we obtain all real

forms24 of (58) which can be expressed as Rosenfeld lines over R⊗O'Oor over R⊗Os'Os;

they are summarized by the following Table :

Plane Isometry Isotropy #nc #cχ

OP1Spin (9) Spin (8) 0 8 −8

OH1Spin (8,1) Spin (8) 8 0 8

OsP1Spin (5,4) Spin (4,4) 4 4 0

along with OsH1'OsP1.

It should then be remarked that the octonionic line OP1can be identiﬁed with the 8-sphere

S8≡S8

Rover R, the hyperbolic line OH1with the 8-hyperboloid H8and the split-octonionic line

OsP1with the Kleinian 8-hyperboloid H4,4(for some applications to physics and to non-compact

versions of Hopf maps, see e.g. [Ha]).

The real forms of OP1

C(58) have the general structures

Spin (p, 9−p)

Spin (p, 8−p)

structure I

or Spin (p, 9−p)

Spin (p−1,9−p)

structure II

,(59)

with p= 0,1, ..., 9, which gets exchanged under p↔9−p.

OP1,OH1and OsP1'OsH1respectively correspond to structure Iwith p= 0,8,4(or

structure II with p= 9,1,5).

All other real forms of OP1

C(58) cannot be characterized as Rosenfeld lines over O(or Os).

6.2 C⊗O

In the bioctonionic case, i.e. for C⊗O, the starting point is given by the bioctonionic projective

line over C, i.e. by the locally symmetric coset manifold having as isometry group Spin(10,C)

with Spin (8,C)⊗U(1) as isotropy group :

(C⊗O)P1

C'Spin (10,C)

Spin (8,C)⊗(U(1))C

.(60)

23Also in this case, this restriction does not imply any loss of generality, as far as the other, non-octonionic

Rosenfeld lines can be obtained as suitable sub-manifolds of the octonionic Rosenfeld lines.

24By real forms of OP1

C, we here mean the cosets with isometry groups given by all real (compact and non-

compact) forms of Spin(9,C), and with isotropy group given by all (compact and non-compact) real forms of

Spin(8,C)which are subgroups of the corresponding real form of Spin(9,C).

19

In this coset space formulation, the tangent space of (C⊗O)P1

Ccan be identiﬁed with the

8(v,s,c),C+⊕8(v,s,c),C−representation of Spin (8,C)⊗(U(1))C, where, as above, d4-triality [Por]

allows to equivalently choose v,s, or c.

Then, by specifying the formulæ (56) and (57) for A=Cor Csand B=Oor Os, we obtain

all real forms25 of (60) which can be expressed as Rosenfeld lines over C(or Cs)⊗O(or Os); they

are summarized by the following Table :

Plane Isometry Isotropy #nc #cχ

(C⊗O)P1Spin (10) Spin (8) ⊗U(1) 0 16 −16

(C⊗O)H1Spin (8,2) Spin (8) ⊗U(1) 16 0 16

(C⊗Os)P1Spin (6,4) Spin (4,4) ⊗U(1) 8 8 0

(Cs⊗O)P1Spin (9,1) Spin (8) ⊗SO (1,1) 8 8 0

(Cs⊗Os)P1Spin (5,5) Spin (4,4) ⊗SO (1,1) 8 8 0

along with

(C⊗Os)P1'(C⊗Os)H1;(61)

(Cs⊗O)P1'(Cs⊗O)H1;(62)

(Cs⊗Os)P1'(Cs⊗Os)H1.(63)

The real forms of (C⊗O)P1

C(60) have the general structures

Spin (p, 10 −p;C)

Spin (p, 8−p;C)⊗U1

structure I

or Spin (p, 10 −p;C)

Spin (p−2,10 −p;C)⊗U1

structure II

,(64)

Spin (p, 10 −p;C)

Spin (p−1,9−p;C)⊗SO (1,1)

structure III

,(65)

Y16 := SO∗(10)

SO∗(8) ⊗U(1) 'SO∗(10)

Spin (6,2) ⊗U(1),(66)

with p= 0,1, ..., 10, such that structures Iand II get exchanged (while structure III is invari-

ant) under p↔10 −p.

The planes (C⊗O)P1,(C⊗O)H1,(C⊗Os)P1respectively correspond to structure Iwith

p= 0,8,6(or structure II with p= 10,2,4), whereas (Cs⊗O)P1and (Cs⊗Os)P1respectively

correspond to structure III with p= 9 (or p= 1) and p= 5. All other real forms of (C⊗O)P1

C

(60), and in particular (66), cannot be characterized as Rosenfeld lines over C(or Cs)⊗O(or Os).

6.3 H⊗O

In the quateroctonionic case, i.e. for H⊗O, the starting point is given by the quateroctonionic

projective line over C, i.e. by the locally symmetric coset manifold having as isometry group

Spin (12,C)with Spin (8,C)⊗Spin(4,C)as isotropy group :

(H⊗O)P1

C'Spin (12,C)

Spin (8,C)⊗Spin (4,C).(67)

25By real forms of (C⊗O)P1

C, we here mean the cosets with isometry groups given by all real (compact and

non-compact) forms of Spin(10,C), and with isotropy group given by all (compact and non-compact) real forms

of Spin(8,C)⊗(U(1))Cwhich are subgroups of the corresponding real form of Spin(10,C).

20

In this coset space formulation, the tangent space of (H⊗O)P1

Ccan be identiﬁed with the

8(v,s,c),2,2representation of Spin(8,C)⊗Spin(4,C), where we used Spin(4,C)'SL(2,C)⊗SL(2,C)

and, as above, d4-triality [Por] allows to equivalently choose v,s, or c.

Then, by specifying the formulæ (56) and (57) for A=Hor Hsand B=Oor Os, we obtain

all real forms26 of (67) which can be expressed as Rosenfeld lines over H(or Hs)⊗O(or Os); they

are summarized by the following Table :

Plane Isometry Isotropy #nc #cχ

(H⊗O)P1Spin (12) Spin (8) ⊗SU (2) ⊗SU (2) 0 32 −32

(H⊗O)H1Spin (8,4) Spin (8) ⊗SU (2) ⊗SU (2) 32 0 32

(H⊗Os)P1Spin (8,4) Spin (4,4) ⊗SU (2) ⊗SU (2) 16 16 0

(Hs⊗O)P1Spin (10,2) Spin (8) ⊗SL (2,R)⊗SL (2,R) 16 16 0

(Hs⊗Os)P1Spin (6,6) Spin (4,4) ⊗SL (2,R)⊗SL (2,R) 16 16 0

along with

(H⊗Os)P1'(H⊗Os)H1;(68)

(Hs⊗O)P1'(Hs⊗O)H1;(69)

(Hs⊗Os)P1'(Hs⊗Os)H1.(70)

The real forms of (H⊗O)P1

C(67) have the general structures

Spin (p, 12 −p;C)

Spin (p, 8−p;C)⊗SU (2)⊗2

structure I

or Spin (p, 12 −p;C)

Spin (p−4,12 −p;C)⊗SU (2)⊗2

1

structure II

,(71)

Spin (p, 12 −p;C)

Spin (p−1,9−p;C)⊗SL(2,C)R

structure II I

or Spin (p, 12 −p;C)

Spin (p−3,11 −p;C)⊗SL(2,C)R

structure IV

,(72)

Spin (p, 12 −p;C)

Spin (p−2,10 −p;C)⊗SL(2,R)⊗2

structure V

,(73)

Y32 := SO∗(12)

SO∗(8) ⊗SO∗(4) 'SO∗(12)

Spin (6,2) ⊗SU (2) ⊗SL (2,R),(74)

with p= 0,1, ..., 12, such that structures Iand II (and structures I II and IV ) get exchanged

(while structure Vis invariant) under p↔12 −p.

(H⊗O)P1,(H⊗O)H1and (H⊗Os)P1respectively correspond to structure Iwith p= 0,

8,4(or structure II with p= 12,4,8), whereas (Hs⊗O)P1and (Hs⊗Os)P1respectively

correspond to structure Vwith p= 2 (or p= 10) and p= 6.

All other real forms of (H⊗O)P1

C(67), and in particular (74), cannot be characterized as

Rosenfeld lines over H(or Hs)⊗O(or Os).

6.4 O⊗O

In the octooctonionic case, i.e. for O⊗O, the starting point is given by the octooctonionic

projective line over C, i.e. by the locally symmetric coset manifold having as isometry group

Spin (16,C)with Spin (8,C)⊗Spin(8,C)as isotropy group :

(O⊗O)P1

C'Spin (12,C)

Spin (8,C)⊗Spin (8,C).(75)

26By real forms of (H⊗O)P1

C, we here mean the cosets with isometry groups given by all real (compact and

non-compact) forms of Spin(12,C), and with isotropy group given by all (compact and non-compact) real forms

of Spin(8,C)⊗Spin(4,C)which are subgroups of the corresponding real form of Spin(12,C).

21

In this coset space formulation, the tangent space of (O⊗O)P1

Ccan be identiﬁed with the

8(v,s,c),8(v,s,c)representation of Spin(8,C)⊗Spin(8,C), where, as above, d4-triality [Por] al-

lows to equivalently choose v,s, or c(in all possible pairs for the two Spin(8,C)factors of the

isotropy group).

Then, by specifying the formulæ (56) and (57) for A=Oor Osand B=Oor Os, we obtain

all real forms27 of (75) which can be expressed as Rosenfeld lines over O(or Os)⊗O(or Os); they

are summarized by the following Table :

Plane Isometry Isotropy #nc #cχ

(O⊗O)P1Spin (16) Spin (8) ⊗Spin (8) 0 64 −64

(O⊗O)H1Spin (8,8) Spin (8) ⊗Spin (8) 64 0 64

(O⊗Os)P1Spin (12,4) Spin (8) ⊗Spin (4,4) 32 32 0

(Os⊗Os)P1Spin (8,8) Spin (4,4) ⊗Spin (4,4) 32 32 0

along with

(O⊗Os)P1'(O⊗Os)H1,(76)

(Os⊗Os)P1'(Os⊗Os)H1.(77)

The real forms of (O⊗O)P1

C(75) have the general structures

Spin (p, 16 −p;C)

Spin (p, 8−p;C)⊗Spin (8,C)

structure I

or Spin (p, 16 −p;C)

Spin (p−8,16 −p;C)⊗Spin (8,C)

structure II

,(78)

Spin (p, 16 −p;C)

Spin (p−1,9−p;C)⊗Spin (1,7; C)

structure III

or Spin (p, 16 −p;C)

Spin (p−7,15 −p;C)⊗Spin (1,7; C)

structure IV

,(79)

Spin (p, 16 −p;C)

Spin (p−2,10 −p;C)⊗Spin (2,6; C)

structure V

or Spin (p, 16 −p;C)

Spin (p−6,14 −p;C)⊗Spin (2,6; C)

structure V I

,(80)

Spin (p, 16 −p;C)

Spin (p−3,11 −p;C)⊗Spin (3,5; C)

structure V II

or Spin (p, 16 −p;C)

Spin (p−5,13 −p;C)⊗Spin (3,5; C)

structure V II I

,(81)

Spin (p, 16 −p;C)

Spin (p−4,12 −p;C)⊗Spin (4,4; C)

structure IX

,(82)

Y64 := SO∗(16)

SO∗(8) ⊗SO∗(8) 'SO∗(16)

Spin (6,2) ⊗Spin (6,2),(83)

with p= 0,1, ..., 16, such that structures Iand II ,III and IV ,Vand V I,V I I and V III get

exchanged (while structure IX is invariant) under p↔16 −p.

The planes (O⊗O)P1,(O⊗O)H1and (O⊗Os)P1respectively correspond to structure I

with p= 0,8,4(or structure II with p= 16,8,12), whereas (Os⊗Os)P1corresponds to

structure IX with p= 8. All other real forms of (O⊗O)P1

C(75), and in particular (83), cannot

be characterized as Rosenfeld lines over O(or Os)⊗O(or Os).

27By real forms of (O⊗O)P1

C, we here mean the cosets with isometry groups given by all real (compact and

non-compact) forms of Spin(16,C), and with isotropy group given by all (compact and non-compact) real forms

of Spin(8,C)⊗Spin(8,C)which are subgroups of the corresponding real form of Spin(16,C).

22

7 Conclusions

In this work, in order to provide a rigorous characterization of Rosenfeld “projective” spaces over

(rank-2) tensor products of (division or split) Hurwitz algebras, we have introduced some “magic”

formulæ (28)-(30) (for planes) and (56)-(57) (for lines), which allowed us to characterize many

Rosenfeld spaces as symmetric (pseudo-)Riemannian cosets, whose isometry and isotropy groups

have Lie algebras which are entries of real forms of the order-3 (Freudenthal-Tits [Ti, Fr65])

Magic Square, or of the order-2 (Barton-Sudbery [BS03]) Magic Square.

For “projective” planes, the application of the “magic” formulæ (28)-(30) to the case in

which at least one of the two Hurwitz algebras in the associated tensor product is given by the

octonions Oor by the split octonions Os, allows us to retrieve all (compact and non-compact)

real forms of the corresponding octonionic Rosenfeld planes, except for a limited numbers of

pseudo-Riemannian symmetric cosets, named X32,I ,X32,II ,X64,I ,X64,I I ,X64,II I ,X128,I and

X128,II , and respectively given by (37), (38), (43), (44), (45), (52) and (53). All such cosets

share a common property : up to some possible rank-1 Lie group factor, their isotropy group

is given by the non-compact Lie group SO∗(N)for N= 10,12,16. The fact that such spaces

are not encompassed by our “magic” formulæ (28)-(30) is ultimately due to the fact that the

corresponding Lie algebra so∗(N)does not occur in any real form of the Magic Square of order

2 (except for the case N= 8, for which it holds the special isomorphism so∗(8) 'so(6,2)).

For “projective” lines, the application of the “magic” formulæ (56)-(57) to the case in which at

least one of the two Hurwitz algebras in the associated tensor product is given by the octonions

Oor by the split octonions Os, allows us to retrieve some (compact and non-compact) real

forms of the corresponding octonionic Rosenfeld lines, but still a number of other real forms of

Rosenfeld lines is left out. Again, there are spaces, such as Y16 , Y32 and Y64 (respectively given

by (66), (74) and (83)) that have their isometry and isotropy groups containing factors SO∗(M)

for M= 4,8,10,12,16; but there are also other pseudo-Riemannian spaces left out, given by

(59), (64)-(65), (71)-(73) and (78)-(82).

The fact that some pseudo-Riemannian symmetric spaces are not covered by the classiﬁcation

yielded by the “magic” formulæ (28)-(30) and (56)-(57), which are related to the entries of the

Magic Squares of order 2 and 3, means that not all real forms of Rosenfeld spaces can be realized

as “projective” lines or planes over (rank 2) tensor products of (division or split) Hurwitz algebras.

If one still believe that Rosenfeld’s approach was right after all, one might want to extend our

“magic” formulæ to involve not only unital, alternative composition algebras, i.e. (division or

split) Hurwitz algebras, but also other algebras which are not unital or alternative. Extensive

work on Magic Squares over ﬂexible composition algebras has been done by Elduque (see e.g.

[El04, El02]), whereas a geometrical framework for these algebras has been recently discussed

in [CZ22] and [CMZ22].

Therefore, after all, Rosenfeld might still be right....we hope to report on this in forthcoming

investigations.

Acknowledgments

The work of D. Corradetti is supported by a grant of the Quantum Gravity Research Institute.

The work of AM is supported by a “Maria Zambrano” distinguished researcher fellowship at

the University of Murcia, ES, ﬁnanced by the European Union within the NextGenerationEU

program.

23

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