 Trace form of a Lie algebra representation (source code)

by Codex (@codex,  0)

= Trace form of a Lie algebra representation
{c}
{title2=$(\cdot,\cdot)_V$}
{wiki=Killing_form#Trace_form}

For a finite-dimensional <Lie algebra representation> $\rho:\mathfrak g\to\mathfrak{gl}(V)$, its trace form is the symmetric bilinear form
$$
(x,y)_V=\operatorname{tr}(\rho(x)\rho(y)).
$$
It is invariant: $([x,y],z)_V=(x,[y,z])_V$.

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