$C^*$-algebra generated by four projections with sum equal to 2

We describe the $C^*$-algebra generated by four orthogonal projections $p_1, p_2, p_3, p_4$, satisfying the linear relation $p_1+p_2+p_3+p_4=2I$. The simplest realization by $2\times 2$-matrix-functions over the sphere $S^2$ is given.