## Representing $G=text{GL}^+(2,mathbf R)$ as the matrix product $G=TH$. If $H=text{SO}(2)$, what is $T$?

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 $H=text{SO}(2)$?

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