quantum field theory - Lax-Pair for principal chiral model

This question concerns Eq. (2.10) of the paper http://arxiv.org/pdf/hep-th/0305116v2.pdf by Bena, Polchinski and Roiban.

In section 2.1 they are showing that the infinite number of conserved quantities for the principal chiral model

$$\begin{equation} L = \frac{1}{2\alpha_0} \mathrm{Tr}(\partial_\mu g^{-1}\partial_\mu g) \end{equation}$$

are given by the fixed-time Wilson lines $U(\infty,t;-\infty,t)$ where

$$\begin{equation} U(x;x_0) = \mathrm{P}\, e^{-\int_{\mathcal{C}}a} \end{equation}$$

and $a$ is a 1-parameter family of flat connections given by Eq. (2.3).

My question is what becomes of the last two terms (i.e. $-a_0a_1 +a_1a_0$) in the second line of Eq. (2.10). Do they cancel? I don't see why the should because the $a$'s are non-commuting (Lie algebra-valued).

Thanks