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On a Lie Algebra, we often work with the adjoint representation

\begin{align*} \text{ad}_u: \mathfrak{g}&\to\mathfrak{g}\\ v&\mapsto [u,v]. \end{align*}

Provided we have some symmetric, positive definite map $A : \mathfrak{g} \to \mathfrak{g}^*$ to define an inner product , what would be the explicit form of the pullback $\text{ad}_u^*$ defined by

$$\text{ad}_u^*(A(v))(w) = A(v)(\text{ad}_u w)?$$

I initially thought that this pullback should be equal to $$\text{ad}_u^* = A \circ \text{ad}_u \circ A^{-1}$$ but after some work it seems that this is only true if the induced metric is left and right invariant.

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