I'm in the process of learning the poste Lecture1-The Paul Levy stochstic area formula but I have some question of this topic. Consider the case of the Levy area $$ S_t=\int_0^t B^1_s dB^2_s-B^2_s dB^1_s, $$ where $B_t=(B^1_t,B^2_t)$, $t \ge 0$, is a two dimensional Brownian motion started at $0$. The $3$-dimensional process $X_t=(B^1_t,B^2_t,S_t)$, is solution of a stochastic differential equation \begin{align*} dX^1_t & =dB^1_t\\ dX^2_t & =dB^2_t\\ dX^3_t & =-X^2_t dB^1_t+X^1_t dB^2_t \end{align*}
I do not understand this claim:
As a consequence $X_t$ is a Markov process with generator $$ L=\frac{1}{2} (X^2+Y^2) $$ where $X,Y$ are the following vector fields \begin{align*} X&=\frac{\partial}{\partial x}-y \frac{\partial}{\partial z}\\ Y&=\frac{\partial}{\partial y}+x \frac{\partial}{\partial z}. \end{align*}
The above system of stochastic differential equations can be written as: \begin{equation} d X_t= \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ -X^2_t &X^2_t \\ \end{pmatrix} \begin{pmatrix} dB^1_t \\ dB^2_t \end{pmatrix} \end{equation} I have confusion to apply the It\^o lemma in the multidimensional case because the the matrix \begin{equation*} \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ -X^2_t &X^2_t \\ \end{pmatrix} \end{equation*} is not square.
Thank you for your compehension.