Explicit 7-torsion in the Tate-Shafarevich groups of genus 2 Jacobians

We describe an algorithm which, on input a genus 2 curve C/Q whose Jacobian J/Q has real multiplication by a quadratic order in which 7 splits, outputs twists of the Klein quartic curve parametrising elliptic curves whose mod 7 Galois representations are isomorphic to a sub-representation of the mod 7 Galois representation attached to J/Q. Applying this algorithm to genus 2 curves of small conductor in families of Bending and Elkies--Kumar we exhibit a number of genus 2 Jacobians whose Tate--Shafarevich groups (unconditionally) contain a non-trivial element of order 7 which is visible in an abelian three-fold.