How many free parameters does a $28 \times 28$ symmetric matrix $M$ have if it is subjected to a constraint
$$M^T L M=L \tag{1}$$ where
$$L=\begin{bmatrix} 0&I_6&0\\I_6&0&0\\0&0& -I_{16}\end{bmatrix} \tag{2}$$
I know that a symmetric matrix has $28 \times 29/2=406$ free parameters and I know that the final answer must be $132$, so the constraint $(1)$ should give $274$ constraints. I am not sure how to get to this number. The only constraint counting I have done is for counting parameters of the classical Lie groups. What is the general procedure for approaching problems of this kind? Also, does the matrix $(2)$ have a name or do the situation I am describing has a name/familiar to mathematicians for some reason? It looks like pseudo-orthogonality but I am not sure.
Extra question: Can you suggest a reference where similar matrix equation constraint counting is done as I am new to this. Some MSE links to other problems of this kind will do too. I have seen this but I think the top answer is too concise for a newbie.
What I tried
Let $$M=\exp(i\theta T) \tag{1}$$ where $\theta \in \mathbb{R}$ is a generator and $T$ is a generator. Now, if $M$ is symmetric, then $T$ is also symmetric so that we can choose $T$ to be of the form
$$T=\begin{bmatrix} t_{11}& t_{12}& t_{13}\\t_{12}^T&t_{22}&t_{23}\\t_{13}^T&t_{23}^T&t_{33}\end{bmatrix} \tag{2}$$ where $t_{12}$ is a $6 \times 6$ block and $t_{13}$ and $t_{23}$ are $6 \times 16$ matrices. Moreover, the condition $$M^T L M=L$$ is equivalent to $$ T^T=-L T L^{-1}\tag{3}$$ upon using $(1)$.
Putting the explicit form of $L$ as in the question and the form of $T$ as in $(2)$ in $(3)$ we get the equations
\begin{align} t_{11}+t_{22}&=0 \\ t_{13}-t_{23}^T&=0\\ \end{align}
The first equation says $t_{11}$ is given in terms of $t_{22}$ which has $36$ components being a $6 \times 6$ block. The second equation determines $t_{23}$ in terms of $t_{13}$ completely which has $96$ components being a $16 \times 6$ block. So I am getting a total of $132$ components. Is this correct? Is there an alternative/better way?
Some context: At 20:20 of this lecture by Ashoke Sen, he describes how $132$ scalar fields can be encoded into a $28 \times 28$ of the kind I described above, and asks the viewer to verify this. This is of paramount importance in describing T duality in string theory; so I guess it is a meaningful example problem to learn counting free parameters from a matrix equation.