A Lie group is both a group and a manifold.  When illustrating Lie groups, one runs into several challenges:

• How do you suggest both the group and manifold structure?  The natural inclination is to draw something that mostly captures geometry.  What can you do to also emphasize algebra?
• Some Lie groups are compact; others are noncompact.  Which should you draw?
• When making drawings in , which Lie groups can you even hope to draw?  There aren’t many that can be faithfully depicted as nice manifolds.  One example is , which is geometrically just the circle .  But one-dimensional examples aren’t quite rich enough to illustrate all the important features.  A two-dimensional example is the group of translations of , but geometrically this group doesn’t look any different from the plane itself; one might be mislead to believe that Lie groups are vector spaces (which, in general, they are not).  Something with curvature seems more appropriate.  In the illustration above, I settled on using the 2-sphere (!) as a cartoon of a Lie group.  Of course, is not a Lie group!  But it captures some important features.  For instance: Lie groups are homogeneous, i.e., they “look the same” at every point.  Also, Lie groups don’t have boundary.  Also, most of them are not vector spaces.  Also, does not look so different (geometrically) from the 3-sphere ; is the 3-sphere.  So, using a 2-sphere as a cartoon of a Lie group is not the worst cartoon in the world.
• How should one depict the exponential map?  The good news is that has a meaning not only on Lie groups, but on any manifold.  So in the case of , the surface exponential map can be used as a surrogate for the Lie algebra/Lie group exponential map.  In particular, it will trace out a great circle, which again captures the fact that Lie groups are nice, highly symmetrical spaces, and the exponential map is a nice, straight curve.
• Alluding to the algebraic question, how does one illustrate the group action?  In the example above, I at least draw two tangent spaces: one at the identity and another at some arbitrary point ; the illustration suggests that the pushforward under the left group action is sort of a rotation of tangent planes.  (Again: doesn’t technically make sense for the 2-sphere, but captures the right idea.). The algebraic question is definitely the weakest point of this illustration.  What else can be done here?
• How can the illustration provide more intuition/motivation for Lie groups?  One way is to think about how a Lie group acts on some other space, say, rotations acting on .  In this case, that idea is communicated by drawing a box acted on by rotations; the exponential map spins the box around a fixed axis with a fixed velocity; an arbitrary path through corresponds to some time-parameterized family of rotations.  (Here, perhaps, it is important to remind the reader that not all Lie groups describe rotations!)
• What about infinite-dimensional Lie groups?  Some ideas are well captured by this two-dimensional cartoon; others are not!