Given a complex Hilbert space H, we study the manifold $$𝒜$$ of algebraic elements in $$Z=ℒ\left(H\right)$$ . We represent $$𝒜$$ as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C*-algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0< r<∞) are real-analytic direct submanifolds of Z. Using the C*-algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine...