Simple and semisimple Lie algebras and codimension growth

We study the exponential growth of the codimensions $c_ n^{L}(B)$ of a finite dimensional Lie algebra $B$ over a field of characteristic zero. In the case when $B$ is semisimple we show that $\lim _{n\to \infty } \sqrt[{n}]{c_ n^{L}(B)}$ exists and, when $F$ is algebraically closed, is equal to the dimension of the largest simple summand of $B$. As a result we characterize central-simplicity: $B$ is central simple if and only if $\dim B = \lim _{n\to \infty } \sqrt[{n}]{c_ n^{L} (B)}$.