What is the significance of the $\mathbb Z_2 \times \mathbb Z_2$ subgroup of $SO(3)$?

Solution 1:

You can get a $Z_2\times Z_2$ embedded in $SO(3)$ also via the Klein subgroup of $S_4$ and observe that $S_4$ is the group of symmetries of the regular tetrahedron.

In the book Spin Geometry by Lawson and Michelsohn, on pages 36-37 the authors discuss the finite Clifford group $F_n$ associated with the Clifford algebra $Cl(n)$, as well as the relation between their representations. Since Clifford algebras and matrix algebras are closely related (see e.g., the table on page 29 there), there is a good chance that one might be able to view the Klein subgroup or a suitable extension thereof as the Clifford group of an appropriate algebra, and thereby obtain a connection between their representations.