For an example, take the special orthogonal groups in even dimension.

B corresponds to the special orthogonal group, SO(2r + 1).

For orthogonal groups there is a similar picture, with a couple of complications.

Together with the identity, it forms the center of the orthogonal group.

We now turn to the action of the orthogonal group on the spinors.

The name comes from the fact that it is (isomorphic to) the special orthogonal group of order 4.

The orthogonal group also forms an interesting example of a Lie group.

Homotopy groups above do not change under covers, so they agree with those of the orthogonal group.

We define the special orthogonal group to be the image of Γ.

A few well-known ones include the orthogonal groups and the unitary groups.