Transitive families of projections in factors of type $II_{1}$

We introduce a notion of transitive family of subspaces relative to a type $II_{1}$ factor, and hence a notion of transitive family of projections in such a factor. We show that whenever $\mathcal{M}$ is a factor of type $II_{1}$ and $\mathcal{M}$ is generated by two self-adjoint elements, then $\mathcal{M}\otimes M_{2}(\mathbb{C} )$ contains a transitive family of $5$projections. Finally, we exhibit a free transitive family of $12$projections that generate a factor of type $II_{1}$.