= Killing form of the general linear Lie algebra
{c}
{title2=$\kappa(X,Y)=2n\operatorname{tr}(XY)-2\operatorname{tr}X\operatorname{tr}Y$}
For the <general linear Lie algebra> of all $n\times n$ matrices over a <field> of <characteristic zero>, write $\operatorname{ad}_X=L_X-R_X$ on the matrix space. The traces of $L_XL_Y$ and $R_XR_Y$ are $n\operatorname{tr}(XY)$, whereas both mixed traces are $\operatorname{tr}X\operatorname{tr}Y$. Expanding gives the displayed <Killing form>. Its radical is the scalar matrices because $\kappa(X,Y)=0$ for every $Y$ forces $nX-\operatorname{tr}(X)I=0$. Thus the full general linear algebra is not semisimple, even though its traceless summand is semisimple for $n\ge2$.
// Target: lie-theory.bigb
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