  The present paper studies Hecke rings derived by the automorphism groups of
certain algebras $L_p$ over the ring of $p$-adic integers. Our previous work
considered the case where $L_p$ is the Heisenberg Lie algebra (of dimension 3)
over the ring of $p$-adic integers. Although this Hecke ring is noncommutative,
we showed that a formal power series with coefficients in this Hecke ring
satisfies an identity similar to the rationality of the Hecke series for $GL_2$
due to E.~Hecke. In the present paper, we establish an analogous result in the
case of the Heisenberg Lie algebra of dimension 5 over the ring of $p$-adic
integers. In this case, our identity is similar to the rationality of the Hecke
series for $GSp_4$, due to G.~Shimura.
