Derivations with large separating subspace

In his famous paper The image of a derivation is contained in the radical, Marc Thomas establishes the (commutative) Singer-Wermer conjecture, showing that derivations from a commutative Banach algebra $A$ to itself must map into the radical. The proof goes via first showing that the separating subspace of a derivation on $A$ must lie in the radical of $A$. In this paper, we exhibit discontinuous derivations on a commutative unital Fréchet algebra $\mathcal{A}$ such that the separating subspace is the whole of $\mathcal{A}$. Thus, the situation on Fréchet algebras is markedly different from that on Banach algebras.