I went through this problem in Lie groups:

i) Prove that $SL_2 (\Bbb R) × SL_2 (\Bbb R)$ is a linear Lie group.

I identified $SL_2 (\Bbb R) × SL_2 (\Bbb R)$ with $\{\begin{pmatrix} A & 0 \\ 0 & D \end{pmatrix} \in GL_4(\Bbb R), (A,D)\in SL_2 (\Bbb R) × SL_2 (\Bbb R) \}$

ii) Prove that $(M, N ) · A = M AN^{−1}$ is a topological group action of $SL_2 (\Bbb R) × SL_2 (\Bbb R)$ on $M_2 (\Bbb R)$ ,that preserves the déterminant.

I applied definitions...

iii) prove that the déterminant on $M_2(\Bbb R)$ is a quadratic form of signature $(2,2)$.


iv) Deduce a un morphism of Lie groups: $φ : SL_2 (\Bbb R) × SL_2 (\Bbb R)/ ± (I_2 , I_2 ) → (SO_{2,2})^\circ$ where $SO_{2,2}^\circ$ is the connected component of the neutral element of $SO_{2,2}$.

This is where I stuck and I need some help. How to use the questions above to solve this one? I have little knowledge about quadratic forms

Thank you for your help


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