Q:

Solving Problem: LMIs and block matrices

I have been reading through this paper (https://ieeexplore.ieee.org/document/7995739) where I am stuck with this particular LMI.  If you are familiar with control theory, the author is trying to find conditions that would satisfy
\begin{equation}
\mathcal{W} \triangleq \dot{V}(\xi) + \vert \vert \xi \vert \vert^2 - \mu \vert \vert \eta \vert \vert^2 \leq 0
\end{equation}
where $V(\xi) = \xi^T \hspace{1mm} \mathbb{Q} \hspace{1mm} \xi $ is the Lyapunov function. 
Substituting for the state dynamics (assuming I have taken the derivative properly), $\dot{\xi}$, we get
\begin{align}
\mathcal{W} &= 2 \hspace{0.1cm} \xi^T \mathbb{Q} \left[\begin{pmatrix}
\mathbb{A}_{\overline{K}} & \overline{B} \overline{K} \tilde{B}\\
\mathcal{O} & \mathbb{A}_{L}
\end{pmatrix} \hspace{0.1cm} \xi + \begin{pmatrix}
D_{11} & D_{12}\\
D_{21} & D_{22}
\end{pmatrix} \hspace{0.1cm} \eta \right] + \vert \vert \xi \vert \vert^2 - \mu \vert \vert \eta \vert \vert^2\\
&= \left(2 \hspace{0.1cm} \xi^T \hspace{0.1cm} \left[\mathbb{Q} \left\{ \begin{pmatrix}
\mathbb{A}_{\overline{K}} & \mathcal{O}\\
\mathcal{O} & \mathbb{A}_{L}
\end{pmatrix} + \begin{pmatrix}
\mathcal{O} & \overline{B} \overline{K} \tilde{B}\\
\mathcal{O} & \mathcal{O}
\end{pmatrix} \right\} \right] \hspace{0.1cm} \xi + \xi^T \hspace{0.1cm} \xi \right)+ 2 \hspace{0.1cm} \xi^T \hspace{0.1cm} \mathbb{Q} \hspace{0.1cm} \begin{pmatrix}
D_{11} & D_{12}\\
D_{21} & D_{22}
\end{pmatrix} \hspace{0.1cm} \eta -  \mu \hspace{0.1cm} \eta^T \hspace{0.1cm} \eta
\end{align}
The final form that the paper skips to and that I am trying to attain is:
\begin{equation}
\mathcal{W} = \begin{pmatrix} 
\xi\\
\eta
\end{pmatrix}^T \hspace{0.1cm} \Pi \hspace{0.1cm} \begin{pmatrix} 
\xi\\
\eta
\end{pmatrix}
\end{equation}
with $\Pi = \begin{pmatrix}
\begin{pmatrix}
\mathbb{A}_{\overline{K}} & \mathcal{O}\\
\mathcal{O} & \mathbb{A}_{L}
\end{pmatrix}^T \mathbb{Q} + \mathbb{Q} \begin{pmatrix}
\mathbb{A}_{\overline{K}} & \mathcal{O}\\
\mathcal{O} & \mathbb{A}_{L}
\end{pmatrix} + \mathbb{I}_{2n} & \mathbb{Q} \begin{pmatrix}
D_{11} & D_{12}\\
D_{21} & D_{22}
\end{pmatrix}\\
\begin{pmatrix}
D_{11} & D_{12}\\
D_{21} & D_{22}
\end{pmatrix}^T \mathbb{Q}^T & -\mu \mathbb{I}_{q}
\end{pmatrix} + \begin{pmatrix}
\mathbb{Z}^{-1} \overline{B} \overline{K}\\
\mathcal{O}\\
\mathcal{O}
\end{pmatrix} \begin{pmatrix}
\mathcal{O} & \tilde{B} & \mathcal{O}
\end{pmatrix} + \begin{pmatrix}
\mathcal{O}\\
\tilde{B}^T\\ 
\mathcal{O}
\end{pmatrix} \begin{pmatrix}
\left(\overline{B} \overline{K}\right)^T \mathbb{Z}^{-1} & \mathcal{O} & \mathcal{O}
\end{pmatrix}$
If you expand further, you'll see that the following needs to be true for them to be equal:

$\mathbb{Q} \begin{pmatrix}
\mathbb{A}_{\overline{K}} & \mathcal{O}\\
\mathcal{O} & \mathbb{A}_{L}
\end{pmatrix} + \mathbb{Q} \begin{pmatrix}
\mathbb{A}_{\overline{K}} & \mathcal{O}\\
\mathcal{O} & \mathbb{A}_{L}
\end{pmatrix}$ is the same as $\begin{pmatrix}
\mathbb{A}_{\overline{K}} & \mathcal{O}\\
\mathcal{O} & \mathbb{A}_{L}
\end{pmatrix}^T \mathbb{Q} + \mathbb{Q} \begin{pmatrix}
\mathbb{A}_{\overline{K}} & \mathcal{O}\\
\mathcal{O} & \mathbb{A}_{L}
\end{pmatrix}$
Given that $\mathbb{Q} = \begin{pmatrix}
\mathbb{Z}^{-1} & \mathcal{O}\\
\mathcal{O} & \mathbb{P}
\end{pmatrix}$, $\mathbb{P}$ is a positive-definite matrix with matrices $\mathbb{Z}$ and $\mathbb{Q}$ being symmetric, $2 \hspace{0.1cm} \mathbb{Q} \hspace{0.1cm} \begin{pmatrix}
\mathcal{O} & \overline{B} \overline{K} \tilde{B}\\
\mathcal{O} & \mathcal{O}
\end{pmatrix}$ is the same as $\begin{pmatrix}
\mathcal{O} & \mathbb{Z}^{-1} \hspace{1mm} \overline{B} \overline{K} \hspace{1mm} \tilde{B}\\
\tilde{B} \hspace{1mm} \left(\overline{B} \overline{K}\right)^T \mathbb{Z}^{-1} &  \mathcal{O}
\end{pmatrix}$
$2 \hspace{0.1cm} \xi^T \hspace{0.1cm} \mathbb{Q} \hspace{0.1cm} \begin{pmatrix}
D_{11} & D_{12}\\
D_{21} & D_{22}
\end{pmatrix} \eta$ is the same as $\eta \hspace{0.1cm} \begin{pmatrix}
D_{11} & D_{12}\\
D_{21} & D_{22}
\end{pmatrix}^T \hspace{0.1cm} \mathbb{Q} \hspace{0.1cm} \xi + \xi^T \hspace{0.1cm} \mathbb{Q} \hspace{0.1cm} \begin{pmatrix}
D_{11} & D_{12}\\
D_{21} & D_{22}
\end{pmatrix} \eta$

The curious thing is the above three relations hold (as far as I know) only when the matrices are symmetric which in turn means the submatrices in the block matrices are symmetric but knowing the structure of these matrices, they are not.  I think there is something silly I am doing and can't figure it out. 
P.S. I tried to be as succinct as I could have been but I can understand if I am missing some info.  Please let me know. 

A:

I assume $\mathcal{W}$ is a scalar.  In such case it can also be written as
$$
\mathcal{W} = \frac{1}{2}\left(\mathcal{W}+\mathcal{W}^\top\right). 
$$
This is common practice when formulating a LMI, since it has the advantage that when you factor out $\begin{bmatrix} \zeta^\top & \mu^\top\end{bmatrix}^\top$ the $\Pi$ matrix is symmetric.  Writing your initial inequality $\mathcal{W} \leq 0$ in terms of $\Pi=\Pi^\top$ allows you to also write it as the LMI $\Pi\preceq 0$.
For example for state feedback of a LTI system you have
$$
\dot{x}=A\,x+B\,u
$$
and you want to find
$$
u=K\,x, \\
V(x)=x^\top P\,x, \\
P=P^\top\succ0, \\
\dot{V}(x)<0\ \forall\,x\neq0. 
$$
Where $\dot{V}(x)$ can be written as
$$
\dot{V}(x)=2\,x^\top P\,(A+B\,K)\,x,
$$
but more commonly is written as
$$
\dot{V}(x)=x^\top\left(P\,(A+B\,K) + (A+B\,K)^\top P\right)\,x. 
$$

