## Note on Malcev algebras.(English)Zbl 0166.04203

A Malcev algebra is a vector spare over a field whose multiplication satisfies $x^2 = 0, \quad (xy)(zx) + (xy \cdot z)x + (yz\cdot x)x + (zx \cdot x) y = 0.$ Relative to any bilinear product, define a ternary composition by $[xyz] = x(yz) - y(xz) + (xy)z. \tag{*}$ The following theorems are proved:
I. If $$M$$ is an algebra over the field $$\Phi$$, $$\operatorname{char} \Phi \ne 2, 3$$, whose multiplication satisfies $$x^2 = 0$$ for all $$x\in M$$, then $$M$$ is a Malcev algebra if and only if every transformation of the form $$z \to [xyz]$$ $$(x, y\in M)$$ is a derivation.
II. Let $$M$$ be a Malcev algebra over the field $$\Phi$$, $$\operatorname{char} \Phi \ne 2$$. Then there is a Lie algebra $$\mathfrak L\supseteq M$$, and a subalgebra $$\mathfrak D\subseteq \mathfrak L$$ such that (1) $$\mathfrak L = M \oplus \mathfrak D$$, (2) $$M$$ is stable under the adjoint action of $$\mathfrak D$$, (3) for $$x, y \in M$$, the projection of $$[x, y]$$ on $$M$$ along is the original (Malcev) product of $$x$$ and $$y$$. The author concludes from the above that a Malcev algebra is a Lie algebra if and only if it is a Lie triple system relative to the ternary product defined by (*).

### MSC:

 17D10 Mal’tsev rings and algebras

### Keywords:

Maltsev algebras; Lie algebra; derivation