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Are there some good examples of non-abelian Lie groups $G$ that are easy to visualize?

The "prototypical" abelian one I've been using so far is $G = S^1$, which works great; its Lie Algebra $\mathfrak g = T_e G$ can easily by visualized as the line tangent to $e$ at $S^1$, and so I can easily draw pictures of the exponential map, the left-invariant vector fields, etc. I was wondering whether there was a similar good non-abelian example that I could keep in mind.

This question is possibly a duplicate; I admit I don't really understand the answer.

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Evan Chen
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    It depends on what you mean by visualizing. But the first nonabelian one are $SO(3)$ and $SU(2)$ which are three dimensional. $SU(2)$ is diffeomorphic to $S^3$ though, so you might sort of think it as a sphere. –  Oct 03 '15 at 12:15
  • So you want the algebra to be easy to visualize, not necessarily the group? – rschwieb Oct 03 '15 at 12:15

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$SO(3,\Bbb R)$ is something we visualize all the time. The associated algebra (which is the set of skew symmetric matrices) has a classical picture as "infinitesimal rotations."

rschwieb
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