Let $G$ be a finite-dimensional real Lie group with identity element $e$, and let $\mathfrak g=T_eG$ be its [Lie algebra](/page/Lie%20Algebra). For each $g\in G$, let
denote the conjugation map, and define the adjoint representation at $g$ by
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\begin{align*}
\operatorname{Ad}_g:=d(C_g)_e:\mathfrak g\to\mathfrak g.
\end{align*}
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For each $X\in\mathfrak g$, let $\operatorname{ad}_X:\mathfrak g\to\mathfrak g$ denote the [linear map](/page/Linear%20Map) $Z\mapsto [X,Z]$, and let the Killing form $B:\mathfrak g\times\mathfrak g\to\mathbb R$ be defined by