Tetrahedron equation and quantum $R$ matrices for $q$-oscillator representations

We review and supplement the recent result by the authors on the reduction of the three dimensional $R$ (3d $R$) satisfying the tetrahedron equation to the quantum $R$ matrices for the $q$-oscillator representations of $U_q(D_{n+1}^{(2)})$, $U_q(A_{2n}^{(2)})$ and $U_q(C_n^{(1)})$. A new formula for the 3d $R$ and a quantum $R$ matrix for $n=1$ are presented and a proof of the irreducibility of the tensor product of the $q$-oscillator representations is detailed.