= Adjoint branching to root sl2 subalgebras in rank two
{title2=$\mathfrak g\downarrow_{\mathfrak{sl}_2(\alpha)}$}
A <root string> of length $\Lambda+1$ gives an irreducible <sl2 Lie algebra> module $R(\Lambda)$ when restricting the <Adjoint representation> to the <sl2 subalgebra associated with a root>. Include also the root's own $R(2)$ and one commuting Cartan direction $R(0)$. For $A_2$ either simple root gives $R(2)\oplus2R(1)\oplus R(0)$. For $B_2$, the long root gives $R(2)\oplus2R(1)\oplus3R(0)$ and the short root gives $3R(2)\oplus R(0)$. For $G_2$, the long root gives $R(2)\oplus4R(1)\oplus3R(0)$ and the short root gives $R(2)\oplus2R(3)\oplus3R(0)$.
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