Anti-commutator of angular momentum operators for arbitrary spin I know the commutator of angular momentum operators are
$$
[J_i,J_j]=\mathrm i\hbar \varepsilon_{ijk}J_k.
$$
For spin-1/2 particles, $J_i=\frac\hbar2\sigma_i$ where $\sigma_i$ are Pauli matrices, and I can compute $\{J_i,J_j\}$ from the algebra of $\sigma_i$'s,
$$
[\sigma_i,\sigma_j]= 2\mathrm i\varepsilon_{ijk}\sigma_{k},
\quad
\{\sigma_i,\sigma_j\}=2\delta_{ij}1 \!\!1_2.
$$
When it comes to higher dimensons, these relations are broken.
I want to know what are the relations in higher dimensions.

This answer shows the case for a fundamental representation of $SU(N)$,
$$
\{t^{A},t^{B}\} = \frac{2N}{d}\delta^{AB}\cdot 1_{d} + d_{ABC}t^{C},\tag{1}
$$
where
$$
\mathrm{Tr}[t^{A}t^{B}] = N\delta_{AB}\quad
d_{ABC} = \frac{1}{N}\mathrm{Tr}[\{t^{A},t^{B}\}t^{C}].\tag{2}
$$
If I substitude Eq.(1) into Eq.(2), I get
$$
d_{ABC}=\frac1N \frac{2N}{d}\frac1N\mathrm{Tr}[t^At^B] \cdot  \mathrm{Tr}[t^C]
+\frac1N d_{ABD}\mathrm{Tr}[t^Dt^C]
$$
where $\mathrm{Tr}[t^C]=0$. Above equation dosen't tell me how to compute $d_{ABC}$,
since $\mathrm {Tr}[t^D t^C]=\delta_{DC}\mathrm{Tr}[1_d]=N\delta_{DC}$.
 A: For the adjoint (3d irrep) of su(2), it is straightforward to compute all anticommutators explicitly,
$$
\{J^a,J^b  \}_{mk}= -\epsilon_{amn}  \epsilon_{bnk} -\epsilon_{bmn}\epsilon_{ank}= 2\delta_{ab}\delta_{mk} -(\delta_{am}\delta_{bk}+\delta_{bm}\delta_{ak}),
$$
in  variant normalization, $(J^a)_{mn}=i\epsilon_{amn}$,  which is the least of your problems.
You see by inspection that, for $a\neq b$, the right-hand side is a symmetric traceless matrix, so not a linear combination of the three generators! Your equation (1) fails.
Keep reading other answers in that question. su(2) irreps are real/pseudoreal, so anomaly coefficients vanish, as your QFT text must have  emphasized.
But I am unaware of a generic shortcut expression for M.
Experiment with higher spin representations, all explicit. Can the r.h.s. of $\{S_x,S_z\}$, traceless,  be represented by a linear combination of the three generators? (No! You may readily compute it for arbitrary spin s, given item 4 of this link, and confirm it is symmetric  with vanishing diagonal elements, but is not $\propto S_x$, the only generator with these properties!)
