SU3 stabilizer inside compact G2 ($\varphi =\omega \wedge dy+\mathrm{Re}\Omega$)

On ${\mathbb{C}}^3\oplus \mathbb{R}$, take $\omega =(i/2){\sum }_{j}d{z}_j\wedge d{\bar{z}}_j$ and $\Omega =d{z}_1\wedge d{z}_2\wedge d{z}_3$. 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 stabilizer of $\varphi$. The complexified seven-dimensional representation restricts as $3\oplus \bar{3}\oplus 1$.