Nilpotency of Lie algebra from structure constants

Question

Consider a given set of structure constants $c_{ij}^k$ defining a (finite dimensional) Lie algebra $\mathfrak{L}$, i.e. $$[e_i,e_j] = \sum_{k=1}^N c_{i,j}^k \, e_k \qquad i,j=1,\ldots,N$$ with $N$ denoting the dimension of $\mathfrak{L}$. In terms of the $c_{i,j}^k$ the adjoint representation of the basis element $e_i$ is the matrix $$ ad(e_i) \; = \; \begin{pmatrix}c_{i,1}^1 & \ldots & c_{i,N}^1\\ \vdots & & \vdots\\ c_{i,1}^N & \ldots & c_{i,N}^N\end{pmatrix} \qquad i=1,\ldots,N $$ Assuming the sequence $$[ad(e_1),[ad(e_2),[\ldots[ad(e_{N-1}),ad(e_N)]]]] = 0$$ vanishes, is it correct to conclude that $\mathfrak{L}$ is nilpotent? (Wikipedia describes the required length of the sequence as "sufficiently large". Is $N$ sufficiently large?)