Monte Carlo update based on sub-lattices

I would like to simulate a 2D classical spin system, whose interactions are only nearest neighbor, using Monte Carlo. I would like to use Metropolis for updating. I have seen that when updating one can choose a lattice site at random, or update sequentially. I was wondering if it is possible to update part of the lattice simultaneously by considering only a sub-lattice. For instance, let the lattice be $L \times L$ with $L$ even. Indexing from zero, call every site whose sum of their $x$ and $y$ coordinates is odd part of sub-lattice $A$, and every other site sub-lattice $B$. Is it acceptable to compute probabilities of spin acceptances for all of $A$ simultaneously and then implement those spin changes simultaneously? Then subsequently for $B$?