Commonly used to define the adjoint bundle: for \(P\) a principal bundle with structure group \(G\in \mathsf{Lie}{\mathsf{Grp}}\) and $\mathsf{Lie}(G) \coloneqq{\mathfrak{g}}\in \mathsf{Lie} \mathsf{Alg} $, since there is a big adjoint action \begin{align*} G\curvearrowright{\mathfrak{g}}: { \operatorname{Ad} }_g(M) = gMg^{-1} \end{align*} yielding a representation of \(G\). Thus one can form the associated bundle \begin{align*} { \operatorname{ad}} P \coloneqq P \overset{\scriptscriptstyle {{ \operatorname{Ad} }} }{\times}{\mathfrak{g}},\qquad (pg, x)\sim (p, { \operatorname{Ad} }_g(x)) \end{align*}