Identifying superalgebras with fixed points under Cartan involution I am making my way through the "Foundations of the $AdS_5 x S^5$ Superstring: Part I" paper by Arutyunov/Frolov 2009 (https://arxiv.org/abs/0901.4937v2) and am hoping someone can help me bridge a logical breaking point.
In the introduction, the authors assert that the type IIB Green-Schwarz superstring in the $AdS_5 x S^5$ background can be written as a non-linear sigma-model with the target space being
$$
\frac{\text{PSU(2,2|4)}}{\text{SO(4,1) x SO(5)}}.
$$
Logically, the next first section is a discussion of the superconformal algebra $\mathfrak{psu}(2,2|4)$. In order to get to this superalgebra, they start by introducing $\mathfrak{sl}(4|4)$. They then identify the superalgebra $\mathfrak{su}(2,2|4)$ with the fixed points $M^* =M$ of $\mathfrak{sl(4|4)}$ under Cartan involution
$$
M^* = -H M^{\dagger} H^{-1},
$$
where M is the 8 x 8 block matrix
$$
M=
\left(\begin{array}{cccc}
m&\theta\\
\eta & n\\
\end{array}\right)
$$
and H is the 8 x 8 matrix
$$
H=
\left(\begin{array}{cccccccc}
1&0&0&0&0&0&0&0\\
0&1&0&0&0&0&0&0\\
0&0&-1&0&0&0&0&0\\
0&0&0&-1&0&0&0&0\\
0&0&0&0&1&0&0&0\\
0&0&0&0&0&1&0&0\\
0&0&0&0&0&0&1&0\\
0&0&0&0&0&0&0&1\\
\end{array}\right).
$$
The point where I'm getting lost is when they identify the superalgebra $\mathfrak{su}(2,2|4)$ with the fixed points $M^* =M$ of $\mathfrak{sl(4|4)}$ under Cartan involution. If anybody could explain the significance of fixed points under Cartan involution and the logic behind this route of studying the superconformal algebra, I would greatly appreciate it.
 A: It is just a reality condition on $M$. This will select a particular signature to your space-time and also impose reality conditions to the spinors. You see that as follows: the matrix $H$ can be written as
$$
H=\begin{pmatrix}\eta^{(2,2)}&0\\0&\eta^{(4,0)}\end{pmatrix}
$$
where $\eta^{(2,2)}$ is a metric with signature $(2,2)$ and $\eta^{(4,0)}$ a metric with signature $(4,0)$. If you write $M$ as
$$
M=\begin{pmatrix}m&\theta\\\eta&m\end{pmatrix}
$$
the reality condition
$$
M^{*}=-HM^{T}H
$$
will constraint $m$ to be a $u(2,2)$ generator, while $n$ to be a $u(4)$ generator, i.e.
$$
m^{*} =-\eta^{(2,2)}m^{T}\eta^{(2,2)},\qquad n^{*} =-\eta^{(4,0)}n^{T}\eta^{(4,0)}
$$
If $M$ is a $gl(4|4)$ generator the generators that survive under this reality condition are
$$
u(2,2)\oplus u(4)=u(1)\oplus u(1)\oplus su(2,2)\oplus su(4)
$$.
Imposing the constraint $gl(4|4)\rightarrow sl(4|4)$ constraint
$$
\text{tr}(m)-\text{tr}(n)=0
$$
killing one $u(1)$. The other $u(1)$ that survive is generated by
$$
\text{tr}(m)+\text{tr}(n)
$$
and is projected out by doing $sl(4|4)\rightarrow psl(4|4)$. With the reality conditions imposed we have $psu(2,2|4)$
Note that $su(2,2)\cong so(2,4)$ which is the correct signature for a conformal transformation in Minkowski space-time $\mathbb{R}^{1,3}$.
The fermionic spinors $\theta$ and $\eta$ are in the fundamental (or anti-fundamental) of $SU(2,2)\oplus SU(4)$ so the reality condition consistent with the $SU(2,2)\oplus SU(4)$ transformations involve appropriate contractions with the metrics $\eta^{(2,2)}$ and $\eta^{(4,0)}$. The reality condition $M^{*}=-HM^{T}H$ does this automatically.
