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Lie algebra exponential covering problem

Lie Groups, Physics, and Geometry by Robert Gilmore (2008) 7.2 "The covering problem" gives some amazing intuition on the subject as usual.

 Table of contents 40
- A single exponential map is not enough to recover a simple Lie group from its algebra link nosplit Lie algebra exponential covering problem 5
- The product of a exponential of the compact algebra with that of the non-compact algebra recovers a simple Lie from its algebra link nosplit Lie algebra exponential covering problem 22

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Existence of the matrix logarithm