what you "probably" mean by the inner product induced by A is:
<u,v> = (Au)^{T}(Av).
if AA^{T} = I, then A^{T} = A^{-1}, so A^{T}A = I, as well. thus:
<u,v> = (Au)^{T}(Av) = (u^{T}A^{T})(Av) = u^{T}(A^{T}A)v = u^{T}v = u.v
what you "probably" mean by the inner product induced by A is:
<u,v> = (Au)^{T}(Av).
if AA^{T} = I, then A^{T} = A^{-1}, so A^{T}A = I, as well. thus:
<u,v> = (Au)^{T}(Av) = (u^{T}A^{T})(Av) = u^{T}(A^{T}A)v = u^{T}v = u.v