Polynomial Invariants
The unitary groups are the automorphisms of two polynomials in real non-commutative variables:
These are easily seen to be the real and imaginary parts of the complex form . The two invariants separately are invariants of O(2n) and Sp(2n, R). Combined they make the invariants of U(n) which is a subgroup of both these groups. The variables must be non-commutative in these invariants otherwise the second polynomial is identically zero.
Read more about this topic: Unitary Group