TITLE: Significance of the following Matrix?
QUESTION [8 upvotes]: 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.

REPLY [4 votes]: 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
$$  [[U,V],W] + [[V,W],U] + [[W,U],V] = 0.  $$