I'm reading chapter 15 of Fulton & Harris. On page 212, they let $\phi$ be the automorphism of $\mathbb{C}^n$ mapping $e_i\mapsto e_j$, $e_j\mapsto -e_i$, and $e_k\mapsto e_k$ for $k\neq i,j.$ Then they claim that $\phi$ induces an automorphism $\text{Ad}(\phi)$ of $\mathfrak{sl}_n(\mathbb{C})$ which, among other things, takes $\mathfrak{h}$ to itself.
I first thought, based on the notation, that $\text{Ad}(\phi)(X)$ should be $[\phi,X]$. But this does not take $\mathfrak{h}$ to itself (and this wouldn't give an automorphism of $\mathfrak{gl}_n(\mathbb{C})$, which they claim as well).
So what is this map $\text{Ad}(\phi)$??
Since apparently my comment answered the question, I'll just make it an answer:
Try $\mathrm{Ad}(\phi)(X) = \phi^{-1} \circ X \circ \phi$ (or depending on convention, $= \phi \circ X \circ \phi^{-1}$) instead.
Upper case "$\mathrm{Ad}$" is often group conjugation, lower case "$\mathrm{ad}$" its "derived version", the Lie algebra commutator -- which you tried, and you rightly saw it cannot be meant here.