Looking at a basis and their transformed.
I have two independently evaluated matrix expressions, one matrix is formed out of pairwise dot products of a transformed basis:

nmatrix=Outer[Simplify[
   TensorExpand[#2 . #1]] &, basisnTransformed, basisnTransformed]  

and the matrix product of the actual transformation matrix R and its transpose:

R . Transpose[R]

both turn out to be equal and Mathematica gives me the correct answer:

nmatrix  == R.Transpose[R]  -> True

which is correct.

Now I want to get a step further and say

orthonormalTransformedConditions = {nmatrix == IdentityMatrix[3]}

So nmatrix or the pairwise dot products of a transformed basis is assumed to be the IdentityMatrix[3} in other words I assume the transformed basis is orthnormal and then show that R is also orthonormal by proving

R.Transpose[R] == IdentityMatrix[3]

but the test:

Simplify[
 Equal[R . Transpose[R], IdentityMatrix[3], 
  Assumptions -> orthonormalTransformedConditions]]-> Flase

Why ?