Let $\mathfrak{g}$ be real Lie algebra of matrices over $\mathbb{C}$ that is closed under conjugate transpose $(\cdot)^*$. Why is it the case that for each $X \in \mathfrak{g}$, $\mathrm{ad}(X^*) = \mathrm{ad}(X)^*$? We are using the inner product $<X,Y> := \mathrm{Re} \mathrm{tr}(XY^*)$ on $\mathfrak{g}$.
• What is $\text{ad}(X)^{\ast}$? If it's the adjoint with respect to some inner product on matrices, which inner product? – Qiaochu Yuan Jul 12 '17 at 21:43
• $<X,Y> := \text{Re} \text{tr}(XY*)$. – nigel Jul 12 '17 at 22:04
• @nigel And $\mathrm{Retr}$ is... ? – José Carlos Santos Jul 12 '17 at 22:09
Just write out the definitions and simplify. The definition of adjoints with respect to an inner product says that the equation you want is equivalent to saying that, for all $Y$ and $Z$, $$\langle\text{ad}(X^*)(Y),Z\rangle=\langle Y,\text{ad}(X)(Z)\rangle.$$ Now use the definition of ad to rewrite this as $$\langle[X^*,Y],Z\rangle=\langle Y,[X,Z]\rangle.$$ Next insert the definition of your inner product to rewrite this as $$\text{Retr}([X^*,Y]\cdot Z^*)=\text{Retr}(Y,[X,Z]^*).$$ Next use the fact that, in a Lie algebra of matrices, the Lie bracket is the ordinary commutator, and remember that the transpose of a product is the product of the transposes in the reverse order, to put the desired equation into the equivalent form $$\text{Retr}(X^*YZ^*-YX^*Z^*)=\text{Retr}(YZ^*X^*-YX^*Z^*).$$ Next, note that Retr must mean real part of the trace (not real part of the transpose, which can't be an inner product because it isn't a scalar). Also remember that the trace is linear, so that your desired equation amounts to $$\text{Retr}(X^*YZ^*)-\text{Retr}(YX^*Z^*)= \text{Retr}(YZ^*X^*)-\text{Retr}(YX^*Z^*).$$ Finally, remember that the trace of a matrix product $AB$ is the same as the trace of $BA$. Apply that with $A=X^*$ and $B=YZ^*$ to see that the first terms on the two sides of your desired equation match. The second terms are identical, so the equation holds.
• I suspect that what makes this question confusing is that, in the equation $\text{ad}(X^*)=\text{ad}(X)^*$, the star on the left side means the conjugate transpose of the matrix $X$whereas the star on the right side means the adjoint with respect to your inner product. – Andreas Blass Jul 12 '17 at 23:23