= SU3 stabilizer inside compact G2
{c}
{title2=$\phi=\omega\wedge dy+\operatorname{Re}\Omega$}
On $\mathbb C^3\oplus\mathbb R$, take $\omega=(i/2)\sum_jdz_j\wedge d\bar z_j$ and $\Omega=dz_1\wedge dz_2\wedge dz_3$. The <special unitary group> $SU(3)$ preserves the <Kähler form> $\omega$ by <unitarity> and preserves $\Omega$ by its unit <determinant>. Fixing $y$ therefore embeds it in the compact <G2 (mathematics)> stabilizer of $\phi$. The complexified seven-dimensional <representation> restricts as $\mathbf3\oplus\bar{\mathbf3}\oplus\mathbf1$.
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