Quadratic identity and real ternary derivations

We consider a mapping $D$ from a unital C$^*$-algebra $mathcal{A}$, containing a distinguished tern, into its iterated dual spaces, $mathcal{A}^{(n)}$ which satisfies the quadratic identity (neither linearity nor continuity is assumed). In the first step we prove that $D$ is additive, and by assuming continuity we conclude that $D$ is a real ternary derivation.