  In this paper, we study the Cassels-Tate pairing on Jacobians of genus two
curves admitting a special type of isogenies called Richelot isogenies. Let
$\phi: J \rightarrow \widehat{J}$ be a Richelot isogeny between two Jacobians
of genus two curves. We give an explicit formula as well as a practical
algorithm to compute the Cassels-Tate pairing on
$\text{Sel}^{\widehat{\phi}}(\widehat{J}) \times
\text{Sel}^{\widehat{\phi}}(\widehat{J})$ where $\widehat{\phi}$ is the dual
isogeny of $\phi$. The formula and algorithm are under the simplifying
assumption that all two torsion points on $J$ are defined over $K$. We also
include a worked example demonstrating we can turn the descent by Richelot
isogeny into a 2-descent via computing the Cassels-Tate pairing.
