On the magic square C*-algebra of size 4

In this paper, we investigate the structure of the magic square C*-algebra $A(4)$ of size 4. We show that a certain twisted crossed product of $A(4)$ is isomorphic to the homogeneous C*-algebra $M_4(C(\mathbb{R}{P}^3))$. Using this result, we show that $A(4)$ is isomorphic to the fixed point algebra of $M_4(C(\mathbb{R}{P}^3))$ by a certain action. From this concrete realization of $A(4)$, we compute the K-groups of $A(4)$ and their generators.