Specifically, let G be a simply-connected Lie group with Lie algebra . Let the map exp: be an exponential map, defining,
The general formula is given by: .
Here ad(A) B = [A,B] The first few terms are well-known:
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There is no expression in closed form.
For a matrix Lie algebra the Lie algebra is the tangent space of the identity I, and the commutator is simply [X,Y] = XY - YX; the exponential map is the standard exponential map of matrices,
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When we solve for Z in eZ = eX eY, we obtain a simpler formula:
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We note first, second, third and fourth order terms are:
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