visualize the embedding $\mathrm{GL}(2,\mathbb{C}) \supset \mathrm{SL}(2,\mathbb{C}) \supset \mathrm{SU}(2)$ in a larger $\mathbb{R}^{n}$?

Question

1. How to visualize the following embedding: $$ \mathrm{GL}(2,\mathbb{C}) \supset \mathrm{SL}(2,\mathbb{C}) \supset \mathrm{SU}(2). \tag{1} $$ They are all Lie groups thus manifolds. By this "visualize the following embedding", I meant that

• the $\mathrm{GL}(2,\mathbb{C})$ is a non-compact 8-dimensional (real) manifold.

• the $\mathrm{SL}(2,\mathbb{C})$ is a non-compact 6-dimensional (real) manifold.

• the $\mathrm{SU}(2)$ is a compact 3-dimensional (real) manifold which is a 3-sphere $S^3$.

Since it is fairly easy to visualize $S^3$, which is a unit sphere on a $\mathbb{R}^4$; then we may start from $S^3$ and go to consider the embedding $$ S^3= \mathrm{SU}(2) \subset \mathrm{SL}(2,\mathbb{C}) \subset \mathrm{GL}(2,\mathbb{C}) . \tag{2} $$

2. Could we put everything in (1) and (2) into a flat Euclidean space with sufficient large dimension? Say in $\mathbb{R}^{8}$ or $\mathbb{R}^{n}$?