  In this paper, we give an explicit formula as well as a practical algorithm
for computing the Cassels-Tate pairing on $\text{Sel}^{2}(J) \times
\text{Sel}^{2}(J)$ where $J$ is the Jacobian variety of a genus two curve under
the assumption that all points in $J[2]$ are $K$-rational. We also give an
explicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow
\text{Br}(K)$ under the same assumption. Finally, we include a worked example
demonstrating we can indeed improve the rank bound given by a 2-descent via
computing the Cassels-Tate pairing.
