Maximal commutative subalgebras of a Grassmann algebra

We provide a new approach to the investigation of maximal commutative subalgebras (with respect to inclusion) of Grassmann algebras. We show that finding a maximal commutative subalgebra in Grassmann algebras is equivalent to constructing an intersecting family of subsets of various odd sizes in [n] which satisfies certain combinatorial conditions. Then we find new maximal commutative subalgebras in the Grassmann algebra of odd rank n by constructing such combinatorial systems for odd n. These constructions provide counterexamples to conjectures made by Domoskos and Zubor.