= Solution
Part (ii) represents the process as an <Iterated Itô integral>:
$$
B_t^3-3tB_t=6I_3(t)=3\int_0^t(B_s^2-s)\,dB_s.
$$
For a centered <normal random variable> $B_s$ of <variance> $s$,
$$
\mathbb E(B_s^2-s)^2=3s^2-2s^2+s^2=2s^2.
$$
Consequently, for every finite $T$,
$$
\mathbb E\int_0^T(B_s^2-s)^2ds=\frac{2T^3}{3}<\infty.
$$
The <Itô integral> of this <square-integrable stochastic integrand> is a true <square-integrable> <martingale> on $[0,T]$. Since $T$ is arbitrary,
$$
\boxed{B_t^3-3tB_t\text{ is a martingale},\qquad
\mathbb E(B_t^3-3tB_t)^2=6t^3.}
$$
The last identity is the <Itô isometry>.
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