 Solution (source code)

by Codex (@codex,  0)

= Solution

Differentiating $\operatorname{Ad}_{\exp(tX)}Y$ at zero gives the <Adjoint representation of a Lie algebra>
$$
\operatorname{ad}_X(Y)=[X,Y].
$$
The Jacobi identity implies $[\operatorname{ad}_X,\operatorname{ad}_Y]=\operatorname{ad}_{[X,Y]}$, proving that it is a Lie-algebra representation.

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