Consider coordinates $z_k=x_k+iy_k (k=1,2,3)$ on $\mathbb C^3$. Consider a 2-form $$ \omega = \sum_{k=1}^3 dx_k \wedge d y_k $$ and a 3-form $$ \rho= \mathrm{Im}(dz_1\wedge dz_2 \wedge dz_3) $$ A surprising claim I encounter somewhere is that
Question 1: $$ SU(3)=\{g\in SO(6) \mid g ~\text{preserves} ~\rho ~\text{and} ~\omega\} $$
How to show this? I only know this might be related to special holonomy stuffs, and my attempt is just to write everything explicitly in terms of matrices, but I believe that people may come up with better approaches. Especially I will appreciated it if you can provide any geometric explanation of this.
Moreover, consider $\mathbb R^7 = \mathbb C^3 \times \mathbb R$ with coordinates $(z_1,z_2,z_3,t)$, and then there is a similar question as follows:
Question 2: $$ SU(3)=\{g\in SO(7) \mid g ~\text{preserves} ~\rho+ \omega dt ~\text{and} ~\partial_t \} $$