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In this paper we consider the integral orthogonal group with respect to thequadratic form of signature $(2,3)$ given by $\left(\begin{smallmatrix}  0 & 1 \\ 1 & 0  \end{smallmatrix}\right) \perp \left(\begin{smallmatrix}  0 & 1 \\ 1 & 0  \end{smallmatrix}\right) \perp (-2N)$ for squarefree $N\in \mathbb{N}$. Theassociated Hecke algebra is commutative and the tensor product of its primarycomponents, which turn out to be polynomial rings over $\mathbb{Z}$ in $2$algebraically independent elements.  The integral orthogonal group is isomorphic to the paramodular group ofdegree $2$ and level $N$, more precisely to its maximal discrete normalextension. The results can be reformulated in the paramodular setting by virtueof an explicit isomorphism. The Hecke algebra of the non-maximal paramodulargroup inside $\mathrm{Sp}(2;\mathbb{Q})$ fails to be commutative if $N> 1$.

The Hecke algebras for the orthogonal group $SO(2,3)$ and the paramodular group of degree $2$