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.


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