Special orthogonal group ($SO(n)$, Rotation group)

Group of rotations of a rigid body.

Like orthogonal group but without reflections. So it is a "special case" of the orthogonal group.

This is a subgroup of both the orthogonal group and the special linear group.

We can reach it by taking the rotations in three directions, e.g. a rotation around the z axis: then we derive and evaluate at 0: $L_z$ therefore represents the infinitesimal rotation.

Note that the exponential map reverses this and gives a finite rotation around the Z axis back from the infinitesimal generator $L_z$:

Repeating the same process for the other directions gives: We have now found 3 linearly independent elements of the Lie algebra, and since $SO(3)$ has dimension 3, we are done.

Based on the $L_x$,$L_y$ and $L_z$ derived at Lie algebra of $SO(3)$ we can calculate the Lie bracket as:

Has $SU(2)$ as a double cover.