Consider the following $d$-dimensional Lie algebra, $$[X_i, X_j] = f_{ij}^k X_k$$ so that the Cartan metric is given by $$g_{ij} = f_{ia}^bf_{jb}^a$$ Now, what I wish to know is whether the Cartan metric is literally the metric over the Lie group (which I understand is also a smooth manifold) ? Also, does the measure $$\int \sqrt{g}~ d^d \sigma$$ (where $\sigma^i$ parametrizes the lie group) qualify as a haar measure of the lie group?
2026-03-26 09:35:37.1774517737
Haar measure from Cartan metric?
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