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:
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 :
Repeating the same process for the other directions gives:
We have now found 3 linearly independent elements of the Lie algebra, and since has dimension 3, we are done.
Based on the , and derived at Lie algebra of we can calculate the Lie bracket as:

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