Now that the motivation for the question has emerged (algebraicity of $M$ inside $G$), here is how to handle it.  Let $G$ be a connected reductive $\mathbf{R}$-group, and $P$ a parabolic $\mathbf{R}$-subgroup of $G$, so $P=L\ltimes U$ for the unipotent radical $U$ of $P$ and Levi subgroup $L \subset P$, so $L=Z_G(S)$ for a maximal split $\mathbf{R}$-torus $S\subset P$. (This $S$ is maximal as a split $\mathbf{R}$-torus in $G$ when $P$ is minimal as a parabolic subgroup, not otherwise.) 

Clearly $P(\mathbf{R})=L(\mathbf{R})\ltimes N$ for $N:=U(\mathbf{R})$. Define $A:=S(\mathbf{R})^0$, so $M$ is designed to be a complement to $A$ in $L(\mathbf{R})$. We seek an algebraic construction of $M$.  The maximal compact subgroup of $S(\mathbf{R})$ is $S[2](\mathbf{R})$, and it is a complement to $A$ in $S(\mathbf{R})$ since $S$ is a split $\mathbf{R}$-torus and $\mathbf{R}^{\times} = \mathbf{R}^{\times}[2] \times (\mathbf{R}^{\times})^0$. 

Let $M \subset L(\mathbf{R})$ be a maximal compact subgroup; typically $M$ is disconnected since $L(\mathbf{R})$ may be disconnected though with finite component group. See Hochschild's book *The Structure of Lie Groups* for the good theory of maximal compact subgroups of Lie groups with finite component group; this applies to the Lie group $\mathcal{G}(\mathbf{R})$ for any linear algebraic $\mathbf{R}$-group $\mathcal{G}$. 
By maximality of $M$ and conjugacy/existence results for all such, $M$ contains every central compact subgroup, such as $S[2](\mathbf{R})$.

The "algebraicity" of the theory of compact Lie groups ensures that $M=H(\mathbf{R})$ for a unique closed $\mathbf{R}$-subgroup $H \subset L$ for which $H^0$ is $\mathbf{R}$-anisotropic reductive and $H(\mathbf{R})$ meets every connected component of $H$. 

Note that the connected reductive $L/S$ is $\mathbf{R}$-anisotropic, so the group $L(\mathbf{R})/S(\mathbf{R})=(L/S)(\mathbf{R})$ is *connected* compact.  We claim that $H$ is an isogeny complement to $S$ in $L$ as algebraic groups and that $M = H(\mathbf{R})$ is an exact complement to $A$ in $L(\mathbf{R})$; i.e., $M \times A \rightarrow L(\mathbf{R})$ is an isomorphism. Certainly $M \cap A = 1$, and $\dim M + \dim A = \dim L(\mathbf{R})$ since $\mathscr{D}(L)$ is an $\mathbf{R}$-anisotropic subgroup of $L$ that is an isogeny complement to the maximal central torus in $L$ (that in turn is contained in every maximal torus of $L$, such as $T$, and contains $S$ by design of $L$).  

Thus, the task is entirely about whether $M$ meets every connected component of $L(\mathbf{R})$.  But it is a general fact (as proved in Hochschild's book) that maximal compact subgroups of Lie groups with finite component group meet every connected component.

For the good robust theory of maximal compact subgroups of Lie groups with finite component group, the book of Hochschild is the only reference I am aware of which provides a complete treatment.  For the algebraicity aspects of compact (possibly disconnected) Lie groups, one has to look elsewhere.