Binary Lie algebra

$BL$-algebra

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

\begin{equation}x^2=J(xy,x,y)=0,\label{eq1}\end{equation}

where

$$J(x,y,z)=(xy)z+(yz)x+(zx)y.$$

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 \eqref{eq1}, also needed is the identity

$$J([(xy)y]x,x,y)=0.$$

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

References

[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