$n$-Ary Mal'tsev Algebras

By analogy with $n$-Lie algebras, which are a natural generalization of Lie algebras to the case of $n$-ary multiplication, we define the concept of an $n$-ary Mal'tsev algerba. It is shown that exceptional algebras of a vector cross product are ternary central simple Mal'tsev algebras, which are not 3-Lie algebras if the characteristic of a ground field is distinct from 2 and 3. The basic result is that every $n$-ary algebra of the vector cross product is an $n$-ary central simple Mal'tsev algebra.