= G2 adjoint branching to a short-root sl2 subalgebra
{c}
For a <short root> $\alpha$ in the <G2 root system>, restriction of the <Adjoint representation> to the <sl2 subalgebra associated with a root> gives
$$
\mathfrak g\cong V(3)^{\oplus2}\oplus V(2)\oplus V(0)^{\oplus3}.
$$
Here $V(n)$ is the irreducible <sl2 Lie algebra> representation of <highest weight> $n$ and dimension $n+1$. Every nonzero <root vector> $x\in\mathfrak g_\alpha$ has $\dim Z_{\mathfrak g}(x)=6$: the <raising operator> has a one-dimensional <kernel> in each of these six irreducible summands.
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