Binary Lie algebra

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A linear algebra $A$ over a field $F$ any two elements of which generate a Lie subalgebra. The class of all binary Lie algebras over a given field $F$ generates a variety which, if the characteristic of $F$ is different from 2, is given by the system of identities




If the characteristic of $F$ is 2 and its cardinal number is not less than 4, the class of binary Lie algebras cannot be defined only by the system of identities (*), also needed is the identity


The tangent algebra of an analytic local alternative loop is a binary Lie algebra and vice versa.


[1] A.I. Mal'tsev, "Analytic loops" Mat. Sb. , 36 (78) : 3 (1955) pp. 569–575 (In Russian)
[2] A.T. Gainov, "Binary Lie algebras of characteristic two" Algebra and Logic , 8 : 5 (1969) pp. 287–297 Algebra i Logika , 8 : 5 (1969) pp. 505–522
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Binary Lie algebra. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.T. Gainov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article