Suppose $V$ is a 3-D vector space over a field $k$ with basis $B=\{v_1,v_2, v_3\}$ and consider the linear operators $x,y,z\in\mathbb{gl}(V)$ whose matrices with respect to $B$ are some 3 by 3 $X, Y$ and $Z$. How would one go about verifying $[x,y],[x,z],[y,z]$ are linear combinations of $x,y,z$ and finding these combinations (i.e check that $x,y,z$ span a 3-dimensional Lie subalgebra $L$ of $\mathbb{gl}(V)$)?
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1You mean, short of calculating the commutators and doing the linear algebra? – Andreas Caranti Apr 08 '15 at 11:21
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Kind of, I mean generally of course because otherwise you'd need to know the matrices involved. – user120246 Apr 08 '15 at 11:24
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Just write the commutators as linear combinations of $x,y,z$, and check the Jacobi identity. This will give conditions on the scalars of the linear combination you are looking for. – Dietrich Burde Apr 08 '15 at 12:09
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So all you need to do is $[x, [y,z]] + [y, [z,x]]+[z,[x,y]]=...=0$. – user120246 Apr 08 '15 at 12:34
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Surely you have to take the bracket operator of the matrices and analyse it to find the linear combinations of each lie bracket of the linear operators $x,y,z$? – user120246 Apr 08 '15 at 12:39
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I do not understand what you mean with "analyse the bracket operator". For example, you could have $[x,y]=z,[x,z]=[y,z]=0$, and the Jacobi identity is satisfied. – Dietrich Burde Apr 08 '15 at 15:29