Anticommutators of the spin-1 representation

[Tony3]

TL;DR

I need to find a closed form of the anticommutators of the Pauli matrices in the spin-1 representation.

The Pauli matrices of the spin-1 representation are given by: $T_{1}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$, $T_{2}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix}$ and $T_{3}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}$. I need to find what $\left\{T_{i},T_{j}\right\}$ is equal to.
Doing some calculations, I found that $\left\{T_{1},T_{2}\right\}=i\begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$, $\left\{T_{1},T_{3}\right\}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & -1 \\ 0 & -1 & 0 \end{pmatrix}$,$\left\{T_{2},T_{3}\right\}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & i \\ 0 & -i & 0 \end{pmatrix}$.
Is there a general relation that I can derive from these special relations? I think that I am close, but I can't quite see it.

 

[A. Neumaier]

What else than the explicit matrices do you expect to get? They are as closed from as you can make it.