Split Lie algebras of order 3

We introduce the class of split Lie algebras of order 3 as the natural generalization of split Lie superalgebras and split Lie algebras. By means of connections of roots, we show that such a split Lie algebra of order 3 is of the form [Formula: see text] with [Formula: see text] a linear subspace of [Formula: see text] and any [Formula: see text] a well-described (split) ideal of [Formula: see text] satisfying [Formula: see text], with [Formula: see text], if [Formula: see text]. Additionally, under certain conditions, the (split) simplicity of the algebra is characterized in terms of the connections of nonzero roots, and a second Wedderburn type theorem for the class of split Lie algebras of order 3 (asserting that [Formula: see text] is the direct sum of the family of its (split) simple ideals) is stated.