Universal C*-algebras generated by doubly non-commuting isometries

We give an explicit injective representation of the universal $\mathrm{C}^\ast$-algebra that is generated by doubly non-commuting isometries. This injectivity allows us to prove that such universal algebras embed naturally into each other and also, when combined with Rieffel's theory of deformation, to show that they are nuclear and to compute their K-theory.