Adjoint representations and Lie Algebras

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I have a very superficial understanding of this subject so apologies in advance for what's probably a stupid question.

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

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

Thanks.
 
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