Lie Algebra π–˜π–”(3)

The elements of π–˜π–”(3) are β€œinfinitesimal generators” of rotations. Their basis elements Lx,Ly,Lz satisfy the following commutator relations:

[ Lx , Ly ] = Lz [ Ly , Lz ] = Lx [ Lz , Lx ] = Ly

Any element of π–˜π–”(3) is a linear combination of this basis, for example:

Ο‰=
+
+

Generating SO(3) from a 3D vector space basis

Exponential map:
-x
+x
-y
+y
-z
+z

Generating quaternions from bivectors

Exponential map:

Generating SU(2) from the Pauli matrices

Exponential map: