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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.

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up vote 1 down vote accepted

Probably what you mean is that a possible Cartan subalgebra is the subspace $\mathfrak h$ of diagonal matrices of that form. That is true.

For each $h\in\mathfrak h$ you can find the map $\operatorname{a}(h):\mathfrak g\to\mathfrak g$. The root spaces and the roots can be found by looking for simultaneous eigenvectors for the maps $\operatorname{ad}(h)$, $h\in\mathfrak h$.

Can you get this far?

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Thank you very much. I think I understand what you mean as to finding the roots and root space. However my main question lies in the determination of a Chevalley basis when the type of root system and its realization in linear lie algebra is given. Very grateful for your help~ – ShinyaSakai Mar 14 '11 at 20:13

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