Lower central series of finite dimensional Lie algebra is stable

Question

Let $\mathfrak{g}$ be a Lie algebra, if $\mathfrak{a},\mathfrak{b}$ are subspaces of $\mathfrak{g}$ we define: $$[\mathfrak{a},\mathfrak{b}]=\mathrm{span}\{[a,b]:a\in\mathfrak{a},b\in\mathfrak{b}\}. \tag{1}$$ The lower central series of $\mathfrak{g}$ is the sequence of ideals defined by: $$\mathfrak{g}^0=\mathfrak{g},$$ $$\mathfrak{g}^i=[\mathfrak{g},\mathfrak{g}^{i-1}], \forall i\in \mathbb{N}.$$ On this MathWorld page, I find that if $\mathfrak{g}$ if finite dimensional, the lower central series of $\mathfrak{g}$ is stable. I think that stable means that: $\exists k\in\mathbb{N}:\mathfrak{g}^n=\mathfrak{g}^k, \forall k\geq n$. Why $\mathrm{dim}(\mathfrak{g})<\infty$ implies that the lower central series is stable? I know that: $\mathfrak{g}^{i+1}\subset\mathfrak{g}^i, \forall i$.

Answer 1

Seems like if the dimension is not infinite, there is no way for there to be an infinite subset of nested subspaces that doesn't accumulate at some $\mathcal{g}^i$. At each iteration, either the dimension decreases, (in which case all future elements of the lower central series have lower dimensions), or it doesn't, in which case no change is made, since finite dimensional vector spaces of the same dimension are isomorphic.