With each orthogeometry (P, ⊥) we associate $${{\mathbb {L}}(P, \bot)}$$ , a complemented modular lattice with involution (CMIL), consisting of all subspaces X and X ⊥ such that dim X < ℵ0, and we study its rôle in decompositions of (P, ⊥) as directed (resp., disjoint) union. We also establish a 1–1 correspondence between ∃-varieties $${\mathcal {V}}$$ of CMILs with $${\mathcal {V}}$$ generated by its finite dimensional members and ‘quasivarieties’ $${\mathcal {G}}$$ of orthogeometries: $${\mathcal {V}}$$ consists of the CMILs representable within some geometry from $${\mathcal {G}}$$ and $${\mathcal {G}}$$ of the (P, ⊥) with $${{\mathbb {L}}(P, \bot) \in {\mathcal {V}}}$$ . Here, $${\mathcal {V}}$$ is recursively axiomatizable if and only if so is $${\mathcal {G}}$$ . It follows that the equational theory of $${\mathcal {V}}$$ is decidable provided that the equational theories of the $${\{{\mathbb {L}}(P, \bot)\, |\, (P, \bot) \in \mathcal {G}, {\rm{dim}} P = n\}}$$ are uniformly decidable.