The Casimir on the $K$-fixed tensor square of the minimal representation of $G_{2(2)}$

We compute the quadratic Casimir on the $K$-fixed subspace of the Hilbert tensor square of the unitary minimal representation of the split real group $G_{2(2)}$. In an explicit algebraic basis of invariant tensors, the Casimir acts by a three-term recurrence. Normalizing this basis yields a symmetri