Exponential growth of Lie algebras of finite global dimension

Let $L$ be a connected finite type graded Lie algebra. If dim $L = \infty$ and gldim $\, L<\infty$, then log index $\, L=\alpha >0$. If, moreover, $\alpha<\infty$, then for some $d$, $\sum_{i=1}^{d-1}$   dim$\, L_{k+i} = e^{k\alpha_k}\,,\,\,$ where $\alpha_k \to$ log index $L$ as $k\to \infty\,.$