Antinilpotent Lie Algebras

The class of antinilpotent Lie algebras closely related to the problem of constructing solutions with constant coefficients for the Yang–Mills equation is considered. A complete description of the antinilpotent Lie algebras is given. A Lie algebra is said to be antinilpotent if any of its nilpotent subalgebras is Abelian. The Yang-Mills equation with coefficients in a Lie algebra $L$ has nontrivial solutions with constant coefficients if and only if the Lie algebra $L$ is not antinilpotent. In Theorem 1, a description of all semisimple real antinilpotent Lie algebras is given. In Theorem 2, the problem of describing the antinilpotent Lie algebras is completely reduced to the case of semisimple Lie algebras (treated in Theorem 1) and solvable Lie algebras. The description of solvable antinilpotent Lie algebras is given in Theorem 3.