Write down the basis of the lie algebra

Question

I have the Lie group $G= \{ X(a), a \in \mathbb{R}\}$ where $$\left( \begin{array}{cc} \cosh(a) & \sinh(a) \\ \sinh(a) & \cosh(a) \end{array} \right)$$

And this thing has a tonne of special properties like it's isomorphic to $SO(2)$ (I think - determinant always 1, rotation matrix etc) and $X^T = X$, and then it has this property where if

$$S = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)$$

Then $XSX^T = S$. And apparently I should be able to just write down the basis of the Lie algebra, so I assume it's obvious without even finding the Lie algebra first. Unfortunately it isn't obvious to me, so if anyone can explain this it'd be hugely appreciated!

If it isn't obvious, I'd be equally interested in knowing that too.