Let $B$ be a standard, one-dimensional Brownian motion. Can I show that $[B \cdot B] = B^2 \cdot [B]$, using the "fundamental identity of stochastic integration", namely that $[H \cdot X, Y] = H \cdot [X, Y]$, where $\cdot$ is stochastic integration and $[U, V]$ is the compensator (aka covariation) of $U$ and $V$?
Attempts at a solution
$[B \cdot B] = [B \cdot B, B \cdot B] = B \cdot [B, B \cdot B] = B \cdot \left(B \cdot [B, B]\right) = B \cdot (B \cdot [B])$
Is $B \cdot (B \cdot [B]) = B^2 \cdot [B]$?
$B^2 \cdot [B] = B^2 \cdot [B, B] = [B^2 \cdot B, B]$
Is $[B^2 \cdot B, B] = [B \cdot B]$?
Did answered my question in his comments, however I'd like to provide an alternative answer based on the textbook I'm reading (Jochen Wengenroth's German textbook "Wahrscheinlichkeitstheorie", de Gruyter 2008), which prompted my original question in the first place (specifically the chain of equations at the top of p. 187).
Lemma 1 According to theorem 9.6.1 ("Covariation and partial integration", p. 181), if $X, Y \in CM^{\text{loc}}(\mathcal{F})$ (i.e. if $X, Y$ are continuous local martingales w.r.t. the filtration $\mathcal{F}$), then $[X,Y]$ is a continuous stochastic process adapted to the filtration $\overline{\mathcal{F}}$ with a.s. bounded variation.
Lemma 2 Theorem 9.1 ("Existence of the Stieltjes integral", p. 173) implies (see the paragraph following the theorem) that any stochastic process $H = (H)_{t \in [0, \infty)}$ that can be uniformly approximated by stochastic step processes (in symbols $H \in \overline{T}(\mathcal{F})$) is Lebesgue-Stieltjes integrable w.r.t. any real-valued stochastic process $X = (X)_{t \in [0, \infty)}$ with a.s. bounded variation.
Lemma 3 According to the paragraph at the bottom of p. 182, if $X$ is a stochastic process with a.s. bounded variation, and if $H$ is a stochastic process that is Lebesgue-Stieltjes integrable w.r.t. $X$, then the Itô integral $H \cdot X$ should be interpreted as a Lebesgue-Stieltjes integral.
Lemma 4 According to theorem 9.2.3 ("[Properties of] the Lebesgue-Stieltjes integral", p. 174), if $X$ is a right-continuous stochastic process with a.s. bounded variation and if $G$ and $H$ are Lebesgue-Stieltjes integrable w.r.t. $X$, then $G$ is Lebesgue-Stieltjes integrable w.r.t. $H \cdot X$ precisely when $GH$ is Lebesgue-Stieltjes integrable w.r.t. $X$. If this is the case, the following holds: $$ G \cdot (H \cdot X) = GH \cdot X $$
Theorem 1 $B \cdot (B \cdot [B]) = B^2 \cdot [B]$.
Proof Denote the natural filtration of $B$ by $\mathcal{F}$. $B$ is a martingale w.r.t. $\mathcal{F}$, hence in particular $B \in CM^{\text{loc}}(\mathcal{F})$. Hence by lemma 1 $[B]$ is a continuous stochastic process adapted to $\overline{\mathcal{F}}$ with a.s. bounded variation. Since $B$ is continuous, it can be shown to belong to $\overline{T}(\mathcal{F})$. Therefore by lemma 2 $B$ is Lebesgue-Stieltjes integrable w.r.t. $[B]$. For the same reason $B^2$ is Lebesgue-Stieltjes integrable w.r.t. $[B]$. Hence by lemma 3 $B \cdot [B]$ and $B^2 \cdot [B]$ should be interpreted as Lebesgue-Stieltjes integrals (and as such are well-defined, as indicated above). Therefore by lemma 4, $B$ is Lebesgue-Stieltjes integrable w.r.t. $B \cdot [B]$, and $B \cdot (B \cdot [B]) = B^2 \cdot [B]$, QED
Using the same argument, theorem 1 can be generalized as follows.
Theorem 2 If $X \in CM^{\text{loc}}(\mathcal{F})$ and if $H, H^2 \in \overline{T}(\mathcal{F})$ (this occurs, for instance, if $H$ is right continuous, or if $H \in \overline{T}(\mathcal{F})$ is uniformly bounded), then $H \cdot (H \cdot [X]) = H^2 \cdot [X]$ (and therefore $[H \cdot X] = H^2 \cdot [X]$).