Visualizing Lie groups.

Solution 1:

Here are a few more examples you can easily "visualize":

1. $SO(3)$ is isometric to the projective space $\mathbb R P^3$, when both are equipped with the standard metrics. Its Lie algebra $(\mathfrak{so}(3),[,])$ is isomorphic to $(\mathbb R^3,\times)$ endowed with the cross product;

2. $SO(4)$ is isometric to $S^3\times S^3/\mathbb Z_2$, the quotient of the product of two $3$-spheres by an involution;

3. $SU(2)$ is isometric to the $3$-sphere $S^3$, when both have the standard metrics. Also the symplectic group $Sp(1)$ is isomorphic to $SU(2)$. As such, $SU(2)$ and $Sp(1)$ are the (universal) double cover of $SO(3)=\mathbb R P^3$.

In particular, all of the above are compact and connected Lie groups.