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Can someone please explain to me why if we have a Lie Group, G with elements g, the adjoint representation of something, eg [tex]g^{-1} A_\mu g[/tex] takes values in the Lie Algebra of G?

Ie. Why does it necessarily mean that [tex][A_\mu,A_\nu]=if_{ijk}A_\sigma[/tex] where f is the structure constant

Thanks.