<s>[INST] Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$ Let $V=\mathbb{C^2}$ be the standard representation of $SL_2(\mathbb{R})$

Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$

I will just consider $SL_2(\mathbb{C})$ since there is a 1:1 correspondence between $SL_2(\mathbb{R})$ and $SL_2(\mathbb{C})$
Weight vectors of $V$ are $v_1$ and $v_{-1}$ with weights 1 and -1 respectively
Weight vectors of $V \otimes V \otimes V$:
Weight 3: $v_1 \otimes v_1 \otimes v_1$
Weight 1: $v_1 \otimes v_1 \otimes v_{-1}$, $v_1 \otimes v_{-1} \otimes v_1$ and $v_{-1} \otimes v_1 \otimes v_1$
Weight -1: $v_1 \otimes v_{-1} \otimes v_{-1}$, $v_{-1}\otimes v_{-1} \otimes v_1$ and $v_{-1} \otimes v_1 \otimes v_{-1}$
Weight -3: $v_{-1} \otimes v_{-1} \otimes v_{-1}$
How do I use this information to find the irreducible representations?
 [/INST] One such decomposition is
$
A=\begin{pmatrix}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\cr -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\cr 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\cr 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\end{pmatrix}$
$ B=  \begin{pmatrix}0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\cr -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\cr 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\cr 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\end{pmatrix}$
$C= \begin{pmatrix}0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\cr 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\cr -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\cr 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\cr 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\end{pmatrix}$
Then you have
$A.A=B.B=C.C= -I(8)$
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