Isomorphism between Gaussian Integer Units and Rotation Matrices Order 4

Theorem
Let $\struct {U_\C, \times}$ be the group of Gaussian integer units under complex multiplication.

Let $\struct {R_4, \times}$ be the group of rotation matrices of order $4$ under modulo addition.

Then $\struct {U_\C, \times}$ and $\struct {R_4, \times}$ are isomorphic algebraic structures.

Proof
Establish the mapping $f: U_C \to R_4$ as follows:

From Isomorphism by Cayley Table, the two Cayley tables can be compared by eye to ascertain that $f$ is an isomorphism:

Cayley Table of Gaussian Integer Units
The Cayley table for $\struct {U_\C, \times}$ is as follows:

Group of Rotation Matrices Order $4$
The Cayley table for $\struct {R_4, \times}$ is as follows: