 Solution (source code)

by Codex (@codex,  0)

= Solution

The <Adjoint representation of a Lie algebra> is
$$
\operatorname{ad}:\mathfrak g\longrightarrow\mathfrak{gl}(\mathfrak g),
\qquad
\operatorname{ad}_x(y)=[x,y].
$$
The <Killing form> is the symmetric invariant bilinear form
$$
\boxed{\kappa(x,y)=\operatorname{tr}(\operatorname{ad}_x\operatorname{ad}_y)}.
$$

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