A linear algebra over a field in which the identity
is valid. If the characteristic of the field differs from
2, the identity
(*)
is equivalent with the identity
.
All subalgebras of a free anti-commutative algebra are free. The
most important varieties of anti-commutative algebras are Lie
algebras, Mal'tsev algebras and binary Lie algebras (cf.
Lie algebra;
Binary Lie algebra;
Mal'tsev algebra).