What is special about SU(2) compared with other SU($N$)?

Question

What is so special about SU(2) compared with other SU($N$) for $N>2$?

I would like to spell out the special properties of SU(2) compared with all other SU($N$) for $N>2$?

For SU(2), we have:

1. pseudoreal representations.
2. We have some representations $k=1,2,3,4,5,\dots$ any integer dimensional representations. But there are not complex conjugate representations. So no $\bar{2},\bar{3},\bar{4},\bar{5},\dots,$ etc.
3. We have no outer automorphism for SU(2).
4. Dynkin diagram has a single dot.

For SU($N$), with $N>2$, we have:

1. complex representations.
2. We have some representations $k=1,N,\frac{N(N-1)}{2},\dots$ some integer dimensional representations. But there are also complex conjugate representations for many of them. (Except the adjoint representations. What else representations have no complex conjugate representations?)
3. We have an outer automorphism for SU($N$) as Out(SU($N$))$=\mathbf{Z}/2$. And I suppose this $\mathbf{Z}/2$ does the complex conjugate of representations.
4. Dynkin diagram has multiple dots connecting by a line.

Are my understanding above precise? Are these statements intertwinedly related? How? What else?

Answer 1

The Lie group $SU(2)$ has as underlying manifold the sphere $S^3$, and none of the $SU(N)$ with $N\gt2$ is diffeomorphic to a sphere.
Moreover $SU(2)$ is the universal covering of $SO(3)$, a 2-sheeted covering $SU(2)\to SO(3)$, which has a very rich geometry when $SU(2)$ is interpreted as the unit sphere of the quaternion algebra $\mathbb H$.
We thus say that $SU(2)$ coincides with the spin group $Spin(3)$, a remarkable coincidence between two classical groups.
But why is $SU(N)$ not a sphere for $N\gt2$ ?
The most elegant proof is that $SU(N)$ is parallelizable (i.e. its tangent bundle is trivial), as are all Lie groups, whereas the only parallelizable spheres are $S^1,S^3,S^7$ and they cannot coincide with $SU(N)$ which for $N\gt 2$ has dimension $N^2-1\gt 7$.
Beware however that this magical proof relies on the difficult enumeration of all parallelizable spheres achieved in the late 1950's by Adams, Bott, Kervaire and Milnor.

Bibliography
An extremely beautiful and quite elementary introduction to the geometry of the morphism $SU(2)\to SO(3)$ can be found in Section 9.3 of Artin's Algebra.