= Adjoint branching from G2 to long-root A2
{title2=$\mathbf{14}\downarrow_{SU(3)}=\mathbf8\oplus\mathbf3\oplus\overline{\mathbf3}$}
The <Adjoint representation> of <G2 (mathematics)> has one <weight> at each of its twelve <roots> and two at zero. Its six long <roots> form the <A2 root system>; together with the two zero <weights> they give the <Adjoint representation> of <SU(3)>. The remaining six short <roots> form the <weight systems> of its defining <representation> and <dual representation>. Hence restriction to the long-root <SU(3)> gives $\mathbf{14}=\mathbf8\oplus\mathbf3\oplus\overline{\mathbf3}$. This is a <branching of a representation>; the complement is invariant under the <subgroup>, not under the full <G2 (mathematics)> action.
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