Exercise 19, § 6

Lie Groups and Lie Algebras, III, 388

Let $G$ be a real Lie group. Show that there exists a neighbourhood $V$ of 0 in $L(G)$ such that, for $x, y$ in $V$ :

\exp(t(x + y)) = \lim_{n \to +\infty} \left( \exp \frac{tx}{n} \cdot \exp \frac{ty}{n} \right)^n

\exp(t^2[x, y]) = \lim_{n \to +\infty} \left( \exp \frac{tx}{n} \cdot \exp \frac{ty}{n} \cdot \exp \frac{-tx}{n} \cdot \exp \frac{-ty}{n} \right)^{n^2}

uniformly for $t \in [0, 1]$ .

Solution

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