I'm working with Varadarajan's book "Lie groups, Lie Algebra and their representations". Specifically on the proof of Ado's theorem page 237.
Ado's theorem : Let $\mathfrak g$ be a Lie algebra over $k$ and $\mathfrak n$ it's nilradical ideal. Then there exists a faithfull finite dimensionnal representation $\rho$ of $\mathfrak g$ such that $\rho(X)$ is nilpotent for all $X \in \mathfrak n$.
The proof is done by induction on the dimension of $\mathfrak g.$ The proof starts like this.
Suppose that $\mathfrak g$ is solvable. We can assume that $\mathfrak n \neq \mathfrak g$ (because then $\mathfrak g$ would be be nilpotent and we know the theorem to be true in that case by a previous result.) Now $[\mathfrak g, \mathfrak g] \subset \mathfrak n$; hence if $\mathfrak a $ is any linear subspace of $\mathfrak g$ such that $\mathfrak a \subset \mathfrak n$ and $\dim \mathfrak a = \dim \mathfrak g - 1 $ then $\mathfrak a$ is an ideal and $[\mathfrak g,\mathfrak a] \subset \mathfrak n$. Choose such an ideal $\mathfrak a$.
I don't understand why such an ideal exists exists. If it did exist then the nilradical would be of codimension $1$ in $\mathfrak g$ since we have
- $a \subset \mathfrak n$
- $\dim \mathfrak a = \dim \mathfrak g - 1$
- $\mathfrak n \neq \mathfrak g$
I find this very odd. Could it be that the author made a mistake and that the text should read $\mathfrak n \subset \mathfrak a$ (I have not found an errata)? This makes for more sense to me. Or a am I missing something ?
