I've this doubt from proof in book of Humphreys Lie Algebras, chapter 2, first theorem: we have a solvable algebra $\;L\le \mathfrak g\mathfrak l(V)\;,\;\;\dim V<\infty\;$ and $\;V\neq0\;$ . Then V contains a common eigenvector for all elements in $\;L\;$ .
Question: I've problems in second parraph of proof: Why is it true that if I have a subspace of $\;L/[LL]\;$ of codimension $\;1\;$ , then its inverse image (under the canonical homomorphism, I think. It isn't said) is ideal of codimension $\;1\;$ , too?
By correspondence theorem (of rings, groups, not the one between Lie groups and algebras), I know index of ideal in $\;L/LL]\;$ is the same as index of inverse image of that ideal in $\;L\;$ , but I am unable to connect this with codimension.
Any help is greatly appreciated.