Consider the group $$\mathfrak{G}=\{g\in M(5,\mathbb{R})\;|\;g^{t}Bg=B,\;\det g=1,\;g_{11}>0\},$$ where $B$ is the matrix $$B=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 \end{array}\right).$$
Now take the subgroup $$\mathfrak{K}=\{u\in\mathfrak{G}\;|\;\sigma u\sigma^{-1}=u\},$$ where $\sigma$ is $$\sigma=\left(\begin{array}{cccc} -1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & -1 \end{array}\right).\\$$
In the paper that I am reading the author states that $\mathfrak{G}/\mathfrak{K}$ is a homogeneous manifold.
To define a topological quotient space one needs an equivalence relation over the original space. In our case, $\sigma$ is not even an element of $\mathfrak{G}$ so I can't find the way to define such a relation on this space using the definition of $\mathfrak{K}$.
Surely I'm missing something here... Could anyone explain why this quotient space is well-defined? In the paper the author uses theory of Lie groups and Lie algebras, which I'm just starting to learn about, and the gap might come from there.
Thank you in advance.