Math question: Let $\phi = \phi_A X^A \in \mathfrak g$ where $\phi_A \in \mathbb C$ for all $A =1,\cdots, \dim \mathfrak g$, and $X^A$ form a basis of some semi-simple Lie algebra $\mathfrak g$ over $\mathbb C$. Here we are viewing $\phi_A$ to be a vector transforming in the adjoint representation of $\mathfrak g$ with $X^A$ being some matrices in the defining representation (that is, matrices which define $\mathfrak g$ as a linear, matrix Lie algebra). Also, by $\phi^{\dagger}$ we mean the complex conjugate transpose of $\phi$. Prove, if $[\phi, \phi^{\dagger}] = 0$, then $\phi$ must be an element of a Cartan subalgebra of $\mathfrak g$, where $[\cdot, \cdot]$ is the standard Lie bracket i.e. commutator on $\mathfrak g$.
Physics context: when studying the moduli space of vacua of $4d$ $\mathcal N=2$ supersymmetric gauge theories which possess a Lagrangian, i.e. $4d$ $\mathcal N=2$ SUSY Yang-Mills theories with some gauge group $\mathfrak g$, the component of the space of vacua called the Coulomb branch is defined (at least classically) by solutions to the equation $[\phi, \phi^{\dagger}]=0$ where $\phi_{A} (\phi)$ is an adjoint (matrix)- valued complex scalar field operator from the $4d$ $\mathcal N=2$ vector multiplet. It should be understood I am abusing notation a bit and should really replace $\phi_A \to \langle \phi_A \rangle \in \mathbb C$ in the above equation to denote the vacuum expectation value of the field operator $\phi_A$, but this is usually understood in physics contexts so that the above equation makes sense. See this OG paper, pg. 7-8 for example.
Solution 1: $[\phi, \phi^{\dagger}]=0$ implies $\phi$ is a normal matrix and therefore is diagonalizable. Then, it can be proven that diagonalizability of $\phi$ as a matrix implies $\phi$ is also ad-diagonalizable, see e.g, 1 or 2. Finally, from this question, we learn that if $\phi$ is ad-diagonalizable i.e. semi-simple, then it is contained in some Cartan subalgebra since, for semi-simple Lie algebras, the union of the CSA's equals the set of semi-simple elements of $\mathfrak g$.
Solution 2: At least when we are talking about the classical matrix Lie algebras $A_n, B_n, C_n, D_n$ over $\mathbb C$, since we can diagonalize $\phi$, it becomes a diagonal matrix. Then, according to this and this answer we see that the set of diagonal matrices so defined in each of these respective Lie algebras forms a Cartan subalgebra, implying $\phi$, once diagonalized, is in this Cartan subalgebra. (For the physicists, we are allowed to diagonalize $\phi$ because this is a gauge transformation.)
These solutions seem a bit awkward/convoluted. Are there solutions which are faster requiring just basic Lie/linear algebra facts?
