 Solution

The space of real $2\times 2$ matrices has a determinant form of signature $(2,2)$: writeThe action $(A,B):X\mapsto AX{B}^{-1}$ of $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ preserves this determinant. Each factor is connected by polar decomposition of an invertible real matrix: its orthogonal factor lies in connected $SO(2)$ and its positive factor is an exponential of a symmetric traceless matrix. Thus the image is contained in $S{O}_0(2,2)$.

If the action is trivial, $X=I$ gives $A=B$, and fixing every $X$ makes $A$ a scalar matrix. Determinant one gives $A=B=\pm I$. The kernel is therefore the diagonal ${\mathbb{Z}}_2$. Its discreteness makes the differential injective; domain and target both have dimension six. An open subgroup of connected $S{O}_0(2,2)$ is the whole group, soAgain the kernel is diagonal, and the target is the identity component.