Heisenberg Hamiltonian matrix and subspaces

In the Ising case (only ZZ terms) you could but now you have to be more careful because you have the XX and YY terms in the Hamiltonian (isotropic Heisenberg model). Take for example the state $| uparrow downarrow downarrow downarrow rangle$ which by your conjecture should be in the $4 times 4$ block of the Hamiltonian. The XX term mixes this state with $| uparrow uparrow uparrow downarrow rangle$, which is outside your $4 times 4$. The correct thing to do is to divide the Hilbert space into blocks of different total spin. As a simpler example consider the two-site case. There you find the famous decomposition $frac{1}{2} otimes frac{1}{2} = 1 oplus 0$ where $1$ is the 3-dimensional triplet sector and $0$ is the 1-dimensional singlet. With 4 sites you get something even more complicated.