Formal problem statement. For a real vector $\vec r$ in bases of $\sigma_i\otimes \sigma_j$ with $i,j=0,1,2,3$, $i$ and $j$ not equal to $0$ at the same time, $\sigma_0=I$ and other indexes stands for Pauli matrices. There are $15$ of them. How to prove that under the action of unitary operation(4 by 4 matrix), i.e. $U\sum_{ij}r_{ij}\sigma_i\otimes \sigma_j U^\dagger$, the length(Euclidean norm) of $\vec r$ remain unchanged(in the prescribed bases)? We can only consider SU(4) elements.
Here are some of my thoughts. I refer to the 'qubit' case for some inspiration, i.e., the bases are only $3$ Pauli matrices, and the vector has $3$ dimensions. A brute-force proof is to check the square of the coefficients of $U\sigma_1U^\dagger$ sum to 1, the square of the coefficients of $U\sigma_2U^\dagger$ sum to 1, and also the square of coefficients of $U\sigma_3U^\dagger$ sum to 1. Then for any term $\sum_{i=1,2,3}r_i\sigma_i$, after the action of unitary, i.e. $U\sum_{i=1,2,3}r_i\sigma_i U^\dagger$, easy to see the length of $\vec r$ remain unchanged. But the method becomes too complicated in the '$2$-qubit' case for we need to check $15$ matrix equations in total. So for the '$2$-qubits' case, is there some easier method, and also, for the '$d$-qubit' case, is this character still remain?