We prove that one can obtain natural bundles of Lie algebras on rank two $s$-Kähler manifolds, whose fibres are isomorphic respectively to $\mathbf{so}(s+1,s+1)$, $\mathbf{su}(s+1,s+1)$ and $\mathbf{sl}(2s+2,ℝ)$. These bundles have natural flat connections, whose flat global sections generalize the Lefschetz operators of Kähler geometry and act naturally on cohomology. As a first application, we build an irreducible representation of a rational form of $\mathbf{su}(s+1,s+1)$ on (rational) Hodge classes of Abelian varieties with rational period matrix.