9 questions linked to/from The anticommutator of $SU(N)$ generators
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### Trace and adjoint representation of $SU(N)$

In the adjoint representation of $SU(N)$, the generators $t^a_G$ are chosen as $$(t^a_G)_{bc}=-if^{abc}$$ The following identity can be found in Taizo Muta's book "Foundations of Quantum ...
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### The conjugate representation in $\mathfrak{su}(2)$

Cheng & Li gives the following problem: Let $\psi_1$ and $\psi_2$ be the bases for the spin-1/2 representation of $\mathfrak{su}(2)$ and that for the diagonal operator $T_3$, \begin{align} ...
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### Is there an anticommutator relation for orbital angular momentum?

So I know that there are commutator relations for $L$ such as $[L_x,L_y] = i\hbar L_z$, but is there a relation for the anticommutator? For example, $L_xL_y + L_yL_x$?
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### Quadratic Casimir operator of higher dimensional $\mathfrak{su}(3)$ representations

In higher dimensional representations of $\mathfrak{su(3)}$, what will be the quadratic Casimir operator? Is it same as in lower dimensions or different?
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### Anticommutator of spin-1 matrices

We know that in the spin-1/2 representation the anticommutation relation of the Pauli matrices is $\{\sigma_{a},\sigma_{b}\}=2\delta_{ab}I$. Does a similar relation hold for the spin-1 representation?
### Product of generators in fundamental representation of $SU(N)$
I'm trying to prove equation 25.20 in Schwartz: $$T^a T^b=\frac{1}{2N}\delta ^{ab}+\frac{1}{2}d^{abc}T^c + \frac{1}{2}if^{abc}T^c,\tag{25.20}$$ where $T^a$ are the fundamental representation ...