4. Given a tetrahedron $ABCD$ satisfying: $AB=2, CD=2\sqrt{5}, AC=BD=3, AD=BC=\sqrt{5}$, then the volume of the tetrahedron is $\qquad$.
Solve:
Establish a spatial rectangular coordinate system $A-xyz$, draw $CH \perp$ plane $ABD$ through $C$. From $AB \perp BC, AB \perp CH$ $\Rightarrow AB \perp BH$, thus $BH // AD$. Then $B(2,0,0), D(0, \sqrt{5}, 0), C(2, y, z)$,
Solve the system of equations $\left\{\begin{array}{l}4+y^{2}+z^{2}=9, \\ 4+(\sqrt{5}-y)^{2}+z^{2}=20,\end{array}\right.$ $\Rightarrow\left\{\begin{array}{l}y=-\frac{3}{\sqrt{5}}, \\ z= \pm \frac{4}{\sqrt{5}} .\end{array}\right.$ Therefore, $V_{ABCD}=\frac{1}{3} \times \frac{1}{2} \times 2 \times \sqrt{5} \times \frac{4}{\sqrt{5}}=\frac{4}{3}$.