 Fourth-moment deficit and bracket variance identity

The polynomial $X^4-6X^2⟨X⟩+3⟨X{⟩}^2$ has zero Itô drift. Finite fourth maximal and second bracket moments make it a true martingale. If $\mathbb{E}{X}_t^2=t$ and $\mathrm{Cov}(X_t^2,⟨X{⟩}_t)=0$, then $\mathbb{E}{X}_t^4=3t^2-3\mathrm{Var}(⟨X{⟩}_t)$. Equality at every time forces a deterministic clock and hence Brownian motion by the Lévy characterization of Brownian motion.