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let ##G## be a Lie group, ##\varrho## its Lie algebra, and consider the adjoint operatores, ##Ad : G \times \varrho \to \varrho##, ##ad: \varrho \times \varrho \to \varrho##.

In a paper (https://aip.scitation.org/doi/full/10.1063/1.4893357) the following formula is used. Let ##g(t)## be a smooth curve on ##G##, with ##\frac{d}{dt}|_{t=0} g(t) = v##. And let ##u## be some arbitrary element in ##\varrho##. Then,

$$\frac{d}{dt}|_{t=0} Ad_{g(t)} u = ad_{\frac{d}{dt}|_{t=0} g(t)} u = ad_{v} u .$$

I know that this identity holds for Matrix groups, but the present setup is a general Lie group.

Furthermore, in the book "Dynamical systems and geometric mechanics", the above property is actually used as a definition of the ##ad##-map, for any ##v \in \varrho## and any curve ##g## with ##g'(0) = v##.

Any hints as to why the formula is true would be greatly appreciated.