  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.
