As is known, $B_2$ can be realized as linear Lie algebra of elements of the form $x= \begin{pmatrix} 0 & b_1 & b_2 \\ c_1 & m & n \\ c_2 & p & q \end{pmatrix}$, where $c_1=-b_2^t$, $c_2=-b_1^t$, $q=-m^t$, $n^t=-n$, $p^t=-p$.
My problem is, how can I find a Chevalley basis in this algebra?
I think $H$ may be the subalgebra whose elements have nontrivial trace entries. It's not difficult for me to find a base for $H$. But the Chevalley constraints turn out very hard to follow. So I think maybe there are some methods for finding Chevalley basis.
I am told that in any semisimple algebra, when a root system $\phi$ and its base $\Delta$ are given, $H$ is determined, so does its base relative to each simple root in $\Delta$. I am eager to know how does this determination goes, for example, in the linear Lie algebra of type $B_2$.
I am not sure whether there are mistakes in my expression as I am not from an English speaking country, but I really appreciate any attention and help. Many thanks.