A question about $\mathfrak{su}(n)$ Let the $\{T^a\}$ be the generators of the adjoint representation of $\mathfrak{su}(n)$, and $f^{abc}$ the structure constants (therefore, $[T^a,T^b]=if^{abc}T^c$ with an implicit sum on the repeated index $c$). I am looking for sets of $n^2-1$ Hermitian traceless matrices $\{X^a\}$ of size $(n^2-1)\times(n^2-1)$ that satisfy
$$[T^a,X^b] = i f^{abc} X^c \tag{1}$$
Obviously, $X^a\equiv const\times T^a$ (with the same constant prefactor for all $a$'s) is a solution of this problem.
The matrices $\{D^a\}$ defined from the symmetric structure constants (sorry for the physicist's jargon; I do not know their proper name - these are the constants that appear, e.g., in the anticommutators of generators in the fundamental representation, $\{t^a,t^b\}=\delta^{ab}/n + d^{abc}t^c$)
by  $(D^a)_{bc}\equiv d^{abc}$ provide another solution.
Does (1) have other solutions (Hermitian traceless), besides linear combinations of the above two?
Is the extra assumption that the $X^a$ should be antisymmetric matrices sufficient for the $\{T^a\}$ to be the unique solution up to a rescaling?
 A: (1) describes tensors transforming under the adjoint representation, Hermitian traceless $(^2−1)×(^2−1)$ dimensional matrices; as a representation the adjoint, your T, is unique, but your Xs need not be a representation, just tensors.  Your T (call them F to comport with standard notation, e.g. Macfarlen et al 1968 ), and D     imaginary antisymmetric and real symmetric, respectively, are very sparse matrices.
But the D s are constructed out of the F, as per equation (2.27) of that reference,
$$
D_i= 2 d_{ijk}F_jF_k ~/n.
$$
The only isotropic tensors of su(n) are, I believe, $f_{ijk}$ and $d_{ijk}$, and you may combine them with the Fs and the Ds to construct adjoint tensors, like your solutions sought. However, bilinears of $f_{ijk}$ and $d_{ijk}$ correlate, equations (2.8-10), and trilinears collapse/reduce, equations (2.15-2.18), so  I have a  sense you will not find new solutions through longer strings of them, but I could be wrong. Recall the quadratic Casimir $F\cdot F$ , the cubic $F\cdot D$, etc..., trivializing the resulting contractions.
Your answer might be in the appendix of  Kaplan and  Resnikoff, Jou  Math  Phys  8 2194 (1967), but it could take work to tease it out...
