Let $\mathcal{A}$ be a Lie algebra. Suppose that adjoint representation of $\mathcal{A}$ is completely reducible (or semisimple). Show that $\mathcal{A}$ can be written as a direct sum of semisimple Lie algebra and abelian Lie algebra.
I know an answer supposing that $\mathcal{A}$ is finite-dimensional Lie algebra. Is there any answer that does not depend on the dimension of algebra?
Thanks!!!