Subgroup structure of special linear group:SL(2,R)

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This article gives specific information, namely, subgroup structure, about a particular group, namely: special linear group:SL(2,R).
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This article describes the subgroup structure of special linear group:SL(2,R), the special linear group of degree two over the field of real numbers.

Subgroups of interest

Subgroup type In what kind of group does this subgroup type notion have significance? To what extent is the subgroup well defined? Value in SL(2,\R)
maximal compact subgroup topological group  ? SO(2,\R) = \left \{ \begin{pmatrix} a & b \\ -b &  a \\\end{pmatrix}\mid a,b \in \R , a^2 + b^2 = 1 \right \} This group is isomorphic to the circle group, and geometrically, it describes rotations.
maximal unipotent subgroup linear algebraic group  ? UT(2,\R) = \left \{ \begin{pmatrix} 1 & x \\ 0 & 1 \\\end{pmatrix} \mid x \in \R \right \}, isomorphic to the additive group of the field of real numbers.
Borel subgroup linear algebraic group  ? \left \{ \begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix} \mid a \in \R^\ast, b \in \R \right \}. It is a semidirect product of the additive group by the multiplicative group where the latter acts by the action of multiplication by the square. It is isomorphic to the direct product of GAPlus(1,R) and cyclic group:Z2.
Weyl group for use in a Bruhat decomposition reductive algebraic group (in the context of a Bruhat decomposition)  ? cyclic group:Z2 as a symmetric group of degree two, embedded inside SL(2,\R) as signed permutation matrices (to be clarified).