Let $V \cong \mathbb{C}^r$ be the defining representation of $\mathfrak{su}(r)$. Then the permutation on $V \otimes V$ can be expressed as

$$ P = \frac{1}{r} \, 1 \otimes 1 + \frac{1}{2} \sum_{a=1}^{r^2-1} t_a \otimes t_a \, , \tag{$\star$} $$

where $\{t_a\}_{a=1}^{r^2-1}$ are a (hermitian, traceless) basis of $\mathfrak{su}(r)$ normalised by $\mathrm{tr}(t_a\,t_b) = 2\,\delta_{ab}$, called generalised Gell-Mann matrices in the physics literature. Special cases of (twice) the $t_a$ are Pauli matrices ($r=2$) and Gell-Mann matrices ($r=3$).

Question: what is the oldest reference for $(\star)$?

I'd think it must be a classical result, but to my surprise the oldest reference that I found is

C. Rakotonirina, Expression of a tensor commutation matrix in terms of the generalized Gell-Mann matrices, Int. J. Math. Math. Sci. 2007:20672 (2007) (arXiv:math/0511451)

NB. The existence of such a formula follows from Schur–Weyl duality. Indeed, since the $t_a$ are orthogonal for the Killing form, $\Omega \propto \sum_a t_a \otimes t_a$ is the split quadratic Casimir of $\mathfrak{su}(r)$ on $V \otimes V$. As $\Omega$ lies in the commutant of the $\mathfrak{su}(r)$, by Schur–Weyl it lies in the image of the $\mathbb{C}S_2$-action, i.e. be a linear combination of the identity and permutation on $V\otimes V$. Solving for the permutation gives an expression of the form $(\star)$.

Let $V \cong \mathbb{C}^r$ be the defining representation of $\mathfrak{su}(r)$. Then the permutation on $V \otimes V$ can be expressed as

$$ P = \frac{1}{r} \, 1 \otimes 1 + \frac{1}{2} \sum_{a=1}^{r^2-1} t_a \otimes t_a \, , \tag{$\star$} $$

where $\{t_a\}_{a=1}^{r^2-1}$ are a (hermitian, traceless) basis of $\mathfrak{su}(r)$ normalised by $\mathrm{tr}(t_a\,t_b) = 2\,\delta_{ab}$, called generalised Gell-Mann matrices in the physics literature. Special cases of (twice) the $t_a$ are Pauli matrices ($r=2$) and Gell-Mann matrices ($r=3$).

Question: what is the oldest reference for $(\star)$?

I'd think it must be a classical result, but to my surprise the oldest reference that I found is

C. Rakotonirina, Expression of a tensor commutation matrix in terms of the generalized Gell-Mann matrices, Int. J. Math. Math. Sci. 2007:20672 (2007) (arXiv:math/0511451)

NB. The existence of such a formula follows from Schur–Weyl duality. Indeed, since the $t_a$ are orthogonal for the Killing form, $\Omega \propto \sum_a t_a \otimes t_a$ is the split quadratic Casimir of $\mathfrak{su}(r)$ on $V \otimes V$. As $\Omega$ lies in the commutant of the $\mathfrak{su}(r)$-action on $V\otimes V$, by Schur–Weyl it lies in the image of the $\mathbb{C}S_2$-action, i.e. is a linear combination of the identity and permutation on $V\otimes V$. Solving for the permutation gives an expression of the form $(\star)$.