In Claire Voisin's book "Hodge Theory and complex algebraic geometry, Vol.1", page 52, there is a very beautiful proof of Newlander-Nirenberg's theorem in real analytic case.
There's one thing about it, though, that bothers me.
Given $U \subset \mathbb{R}^{2n}$ viewed as a real analytic manifold (her goal is to show the thesis locally, so you may think of $(U,\phi)$ on a real analytic manifold $M$), and $I$ almost complex structure on $U$, she says that $I$ can be realized as a
$I:U \to \operatorname{End}\mathbb{R}^{2n}$, such that $I \circ I = -\mathrm{id}$.
She then extends this $I$ to $\mathbb{C}^{2n}$ using a power series argument.
Now, I think she means that we have a real analytic map $$J:U \to \operatorname{End}\mathbb{R}^{2n}$$ that, given a point $u \in U$, gives back $J(u)=I_u$, where $I_u$ is the endomorphism of $T_u U$ given by the almost complex structure. Since in our case $\operatorname{End}T_u U = \operatorname{End}\mathbb{R}^{2n}$ this would be consistent with the rest of the proof.
This makes sense, but I really don't get why she would choose the notation "$J \circ J = -\mathrm{id}$" . It's wrong, the composition of $J$ with itself isn't even well-defined (is it?)... and note that she repeats this notation again later.
So, is there something I don't get about this $I$?
Thanks in advance
First, Claire Voisin is a lady, thus, she should be referred to as "she", not "he". Secondly, in the definition, you do not use the composition $J\circ J=-1$ but matrix product $J(x)^2=-1$ for every $x\in U$. Then, it all makes sense.