Hain's commutator formula for iterated integrals Exercise 7 in Hain's notes on the de Rham fundamental group of $\mathbb{P}^1\setminus \left\{0,1,\infty\right\}$ says the following (for a real or complex smooth manifold $M$):
Let $\alpha, \beta$ be two loops based at a point $x\in M$ and $\omega_1, \omega_2\in \Omega^1  (M)$ two smooth (real or complex) 1-forms on $M$. Then the commutator in $\pi_1 (M,x)$ is related to a period matrix via
$$\int_{[\alpha,\beta]} \omega_1 \omega_2 = \det\left( \matrix{{\int_\alpha \omega_1,\int_\alpha \omega_2}\\{\int_\beta \omega_1 ,\int_\beta \omega_2}}\right).$$ 
I am struggling to prove this using the shuffle product/coproduct/path reversal formulas. Assuming I haven't made a calculation mistake, however, I believe it reduces to showing that
$$\int_\alpha \omega_1\omega_2 + \int_\beta \omega_1\omega_2 + \int_\alpha \omega_2\omega_1 + \int_\beta\omega_2\omega_1 = 2\int_\alpha\omega_1\omega_1+2\int_\beta\omega_2\omega_2$$
i.e.
$$ \left(\int_\alpha\omega_1\right)\left(\int_\alpha\omega_2\right)+ \left(\int_\beta\omega_1\right)\left(\int_\beta\omega_2\right)= \left(\int_\alpha\omega_1\right)^2+\left(\int_\beta\omega_2\right)^2,$$
which I am not totally confident is correct. I have three questions relating to this:
1) Can anybody provide a proof for this formula? I can't find one referenced anywhere, but I believe it is just a clever applyication the basic shuffle product (etc...) formulas for iterated integrals.
2) More conceptually, is there a reason we expect such a relation to hold (i.e. without having to explicitly calculate it)? For example, suppose $M$  has a $2\times 2$ period matrix $A$; as far as I understand this matrix provides the coefficients for an isomorphism between the de Rham cohomology and singular cohomology of $M$. Why should its determinant relate to the iterated integral over the commutator of the two generators of $H_1(M)$ given by the images of $\alpha, \beta$ in the abelianistion map $\pi_1 (M,x)\to H_1 (M)$?
3) When the period matrix is larger, are there further relations between its determinant and more complicated iterated integrals involving elements of $\pi_1 (M,x)$ and the basis elements for $H^1_{ dR} (M)$?
 A: Repeatedly using $\int_{\gamma_1\gamma_2}\omega_1\omega_2=\int_{\gamma_1}\omega_1\omega_2+\int_{\gamma_2}\omega_1\omega_2+\int_{\gamma_1}\omega_1\int_{\gamma_2}\omega_2$ and $\int_{\gamma^{-1}}\omega_1\omega_2=\int_\gamma\omega_2\omega_1$ and $\int_{\gamma^{-1}}\omega=-\int_\gamma\omega$ (all from your lecture notes) you get
$$\int_{\alpha \beta \alpha^{-1} \beta^{-1}} \omega_1 \omega_2=\int_\alpha\omega_1\omega_2+\int_\alpha\omega_2\omega_1+\int_\beta\omega_1\omega_2+\int_\beta\omega_2\omega_1+\int_\alpha\omega_1\int_\beta\omega_2-\int_\alpha\omega_1\int_\alpha\omega_2$$
$$-\int_\alpha\omega_1\int_\beta\omega_2-\int_\beta\omega_1\int_\alpha\omega_2 -\int_\beta\omega_1\int_\beta\omega_2+\int_\alpha\omega_1\int_\beta\omega_2$$
Now use $\int_\alpha\omega_1\int_\alpha\omega_2=\int_\alpha\omega_1\omega_2+\int_\alpha\omega_2\omega_1$ (and the same for $\beta$) to see how all terms cancel  except for 
$$\int_\alpha\omega_1\int_\beta\omega_2-\int_\beta\omega_1\int_\alpha\omega_2$$
as you wanted to show.
I don't have a good answer to your other questions, but have a look at some more complete text about Chen's iterated integrals (and his version of rational homotopy theory) - the best source might by Chen's paper (Iterated path integrals), but there is also a nice old book by Hain (Iterated integrals and homotopy periods).
