TITLE: Equivalent condition for a Lie algebra to be Nilpotent
QUESTION [2 upvotes]: In these notes, the author, N. Perrin, states that a Lie algebra $\frak{g}$ is nilpotent if and only if we can find a decreasing sequence $(\frak{g}_i)_{i=0}^n$ of ideals in $\frak{g}$ such that $\dim\frak{g}_i/\frak{g}_{i+1}=1$ and $[\frak{g},\frak{g}_i]\subset\frak{g}_{i+1}$ for all $i$.
I cannot make any sense of his proof, so I tried proving this on my own. The converse is obvious. When I tried the other implication, I started from the central sequence $(\frak{g}_i)_{i=0}^n$ and said that, if $\frak{g}_i/\frak{g}_{i+1}$ is d-dimensional, then we can find a basis with $d$ elements. Picking a representative for each element of that basis, we get that $\mathfrak{g}_i=\mathbb{C}x_1+\dots+\mathbb{C}x_d+\mathfrak{g}_{i+1}$. So I figured that I could interpolate $\mathfrak{h}_k=\mathbb{C}x_k+\dots+\mathbb{C}x_d+\mathfrak{g}_{i+1}$ between $\frak{g}_i,g_{i+1}$ so the quotients are one-dimensional, but I cannot prove that $\mathfrak{h}_k$ are ideals in all of $\mathfrak{g}$.
I abandoned this idea. Later on, I tried proving that a nilpotent Lie algebra has an ideal of codimension one. I figured that an iteration of this process would yield the result, but apparently I cannot get that the ideals that come from the 2nd application and on of this process are ideals in $\frak{g}$. I am stuck, any help is appreciated.

REPLY [0 votes]: Eventually what I was trying worked after all:
It suffices to interpolate $\mathfrak{h}_k=\mathbb{C}x_k+\dots+\mathbb{C}x_d+\mathfrak{g}_{i+1}$ between $\frak g_i, g_{i+1}$. By the definition of the central series, it is $[\mathfrak{g},\mathfrak{g}_i]=\mathfrak{g}_{i+1}$. Therefore, we have
$$[\mathfrak{g},\mathfrak{h}_k]\subset[\mathfrak{g},\mathfrak{g}_i]=\mathfrak{g}_{i+1}\subset\mathfrak{h}_{k+1}$$
for all $k$, so the desired inclusions hold. Moreover, $\mathfrak{h}_k$ are ideals in $\mathfrak{g}$ and the quotients $\mathfrak{h}_k/\mathfrak{h}_{k+1}$ are obviously one-dimensional.
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