= Solution
By the tip correspondence in the conjugacy,
$$
\xi_t^*=\Phi_t(\xi_t),\qquad \xi_t=\sqrt6\,B_t.
$$
The time-dependent <Itô formula> is valid after localization away from the pole and the terminal time $S$. The map's time variation is of <finite variation>, so there is no extra stochastic cross-variation term. Using the supplied conjugacy <derivative>,
$$
\begin{aligned}
d\xi_t^*&=\Phi_t'(\xi_t)\sqrt6\,dB_t
+\left(\dot\Phi_t(\xi_t)+3\Phi_t''(\xi_t)\right)dt\\
&=\sqrt6\,\Phi_t'(\xi_t)dB_t.
\end{aligned}
$$
Consequently
$$
\boxed{\xi_t^*\text{ is a continuous local martingale on }[0,S),\qquad
[\xi^*]_t=6u(t).}
$$
The cancellation is special to $\kappa=6$: for a general <SLE> parameter the drift would be $(\kappa/2-3)\Phi_t''(\xi_t)dt$. This is the <transformed SLE driving function>, and explains the special <Locality property of SLE> at six.
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