I am looking for rotation invariant scalar functions $f(x,y): x,y \in R^3$ that are **not** some scalar function over the dot product (or norm), i.e. $ f \neq g(x\cdot y, \Vert x \Vert, \Vert y \Vert ) $

Do they exist ?

**Edit:**
Edited to clarify that the norm is just another form of the dot product, and the norm being rotation invariant is really just the dot product being invariant.