Exercise
Let
be a complex finite dimensional Lie algebra and
a nilpotent subalgebra of
. Regard
as an
-module.
Recall that is the generalized weightspace with respect to the weight , so for an element , if and only if for every , there is a positive integer such that .
To prove (1), let and . Then . Hence . Since is finite-dimensional and nilpotent, there is a positive integer such that . Hence Thus , as desired.
To prove (2), note that by definition, is an ideal of if and only if , where . Since by (1), we have . Hence is an ideal of if and only if , as desired.
For (3), choose to be the unique -dimensional complex Lie algebra with a basis such that . Then is a nilpotent subalgebra of such that . Hence is not a Cartan subalgebra, and we have by (1).