Algebras with one of the following identities are considered:
*20c [ [ t1,t2 ],t3 ] + [ [ t2,t3 ],t1 ] + [ [ t3,t1 ],t2 ] = 0, [ t1,t2 ]t3 + [ t2,t3 ]t1 + [ t3,t1 ]t2 = 0, { [ t1,t2 ],t3 } + { [ t2,t3 ],t1 } + { [ t3,t1 ],t2 } = 0, \begin{array}{*{20}{c}} {\left[ {\left[ {{t_1},\;{t_2}} \right],\;{t_3}} \right] + \left[ {\left[ {{t_2},\;{t_3}} \right],\;{t_1}} \right] + \left[ {\left[ {{t_3},\;{t_1}} \right],\;{t_2}} \right] = 0,} \\ {\left[ {{t_1},\;{t_2}} \right]{t_3} + \left[ {{t_2},\;{t_3}} \right]{t_1} + \left[ {{t_3},\;{t_1}} \right]{t_2} = 0,} \\ {\left\{ {\left[ {{t_1},\;{t_2}} \right],\;{t_3}} \right\} + \left\{ {\left[ {{t_2},\;{t_3}} \right],\;{t_1}} \right\} + \left\{ {\left[ {{t_3},\;{t_1}} \right],\;{t_2}} \right\} = 0,} \\ \end{array}
where [t
1
, t
2] = t
1
t
2
− t
2
t
1 and {t
1, t
2} = t
1
t
2 + t
2
t
1
. We prove that any algebra with a skew-symmetric identity of degree 3 is isomorphic or anti-isomorphic to one of such algebras
or can be obtained as their q-commutator algebras.