Significance of the following Matrix? I am unfamiliar with advanced Matrix theory (nor am I a mathematician), so please bear with me.
Is there anything significant about the following Matrix structure? Are there any special symmetries or conserved quantities that can be extracted from this?
$$ \pmatrix{-u_2 & 0 & -\sqrt{2} u_1 & 0 \\
0 & u_2 & 0 & -\sqrt{2} u_1 \\
\sqrt{2} u_1 & 0 & 0 & 0 \\
0 & \sqrt{2} u_1 & 0 & 0} $$
where $u_1,u_2$ are real and have a maximum value of $1$.
Edit:
I was reading something about the correspondence between the group of traceless matrices and the SU(2) group (are they isomorphic?). Am I on the right track? Anything about the blocks that stand out? I was also reading about trace conserving "volume", but I am not sure what that means.
 A: To complile some of the obvious results (which hopefully don't contain mistakes):
It has determinant $4(u_1)^4$. So, it's going to be nonsingular when $u_1>0$. Even when $u_1=0$, as long as it's nonzero, it's still not nilpotent.
If $\lambda$ is an eigenvalue, then the  rest are $\overline{\lambda}$, $-\lambda$ and $-\overline{\lambda}$.
It has trace $0$, whence Lie algebra singles it out as an "infinitesimal volume preserving transformation". (That last point requires a bit of background in Lie algebra to understand.)  It can be reexpressed as $AB-BA$ for two other matrices $A$ and $B$.
This matrix is always diagonalizable.
It is close but not quite anti-symmetric if $u_1>0$. It is easy to see the symmetric part is in the upper left $2\times 2$ block, and the antisymmetric part is everything else.
I started on the singular values, but confirmed in WolframAlpha that they are complicated.
WolframAlpha also confirms that the QR, LU and SVD decmopositions are complicated.
A: You do get a Lie algebra if you ignore the bounds you briefly mentioned. First, define
$$ L \; = \; \left( \begin{array}{cccc} 
1 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1      
\end{array} \right).  $$
and
$$ R \; = \; \left( \begin{array}{cccc} 
0 & 0 & 0 & 1 \\ 
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0      
\end{array} \right).  $$
Next, define a Lie bracket for two of your matrices $V,W$ as
$$ [V,W ] = L (V W - W V) R.  $$ 
This gives a Lie algebra, as the various closure properties hold (addition, multiplication by a scalar) as well as anticommutativity and the Jacobi identity
$$  [[U,V],W] + [[V,W],U] + [[W,U],V] = 0.  $$
