Left SU(2) subgroup of SO(4)
= Left SU(2) subgroup of SO(4)
{title2=$SU(2)_L\subset SO(4)$}
Identify $\mathbb R^4$ with the <quaternions>. Unit <quaternions> act by left multiplication, embedding the <SU(2) group> in $SO(4)$. Under this subgroup, the complexified vector representation branches as $\mathbf4\to2\mathbf2$, and the <Adjoint representation> as $\mathbf6\to\mathbf3\oplus3\mathbf1$. Right multiplication supplies the commuting second factor.