In this paper (Equation 2.6 and 2.7) the author seems to suggest that one can represent the $\text{GL}^+(4,\mathbf R)$ group using the product of two exponentials: $\exp (\epsilon \cdot T) \exp (u \cdot J)$, where $T$ are the generators of shears and dilation, and $J$ are the generators of Lorentz transformations.
My take on the subject is that since $T$ and $J$ do not commute, one cannot write $G$ as a product of these two exponentials. One must instead write $G=\exp ( \epsilon \cdot T + u \cdot J )$. It appears to me the author is wrong.
Is the author correct, or am I?
How can I represent $\text{GL}^+(2,\mathbf R)$ as the matrix product $G=TH$ where $H=\text{SO}(2)$?