= Solution
The space of real $2\times2$ <matrices> has a <determinant> form of signature $(2,2)$: write
$$
X=\begin{pmatrix}u+x&y+v\\y-v&u-x\end{pmatrix},
\qquad \det X=u^2+v^2-x^2-y^2.
$$
The action $(A,B):X\mapsto AXB^{-1}$ 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 $SO_0(2,2)$.
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 $SO_0(2,2)$ is the whole group, so
$$
\boxed{SO_0(2,2)\cong\bigl(SL(2,\mathbb R)\times SL(2,\mathbb R)\bigr)/\mathbb Z_2.}
$$
Again the kernel is diagonal, and the target is the <identity component>.
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