This is a Disphenoid!

The vertices of this tetrahedron are $A\;(0,1,0)$ , $B\;(2,1,0)$ , $C\;(1,0,\sqrt{2})$ and $D\;(1,2,\sqrt{2})$ . Thus the volume of the tetrahedron is the modulus of $\tfrac16 \overrightarrow{AD} \cdot \left(\overrightarrow{AB} \times \overrightarrow{AC}\right) \; =\; \frac16\left(\begin{array}{c} 1 \\ 1 \\ \sqrt{2}\end{array}\right) \cdot \left(\left(\begin{array}{c} 2 \\ 0 \\ 0 \end{array}\right) \times \left(\begin{array}{c} 1 \\ -1 \\ \sqrt{2} \end{array}\right) \right) \; = \; \frac16 \left(\begin{array}{c} 1 \\ 1 \\ \sqrt{2}\end{array}\right) \cdot \left(\begin{array}{c} 0 \\ -2\sqrt{2} \\ -2 \end{array}\right) \; = \; -\frac{2\sqrt{2}}{3}$ and hence is $\boxed{\frac{2\sqrt{2}}{3}}$ Alternatively, this solid is simply a regular tetrahedron of side length $2$ , from which the result about the volume is immediate.