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

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