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Let V and W be subalgebras of a Lie algebra $\mathcal{L}$. I want to show that $[V,W]$ is not always a subalgebra of $\mathcal{L}$.

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    I cannot understand what you are asking.2012-11-27
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    Let V and W be subalgebras of $\mathcal{L}$ a Lie algebra. I want to show that $[V,W]$ is not always a subalgebra of $\mathcal{L}$. Is it clearer? Thank you very much2012-11-27
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    Edit the question and add the explanation there.2012-11-27
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    I made a few changes in your question :D2012-11-27
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    Yes I saw that thanks!2012-11-27

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  • Consider the Lie algebra $\mathcal L=\{X\in M_4(\mathbb R)\,|\,X^T=-X\}$ with the product $[X,Y]=XY-YX$. A basis of this algebra is given by the matrices $u_{i,j}=e_{i,j}-e_{j,i}$ for $i

  • Define the subalgebras $$V=Span(u_{1,2},u_{1,3},u_{2,3})\quad\text{and}\quad W=Span(u_{2,3},u_{2,4},u_{3,4})\,.$$ Then $$[V,W]=Span(u_{1,2},u_{1,3},u_{1,4,},u_{2,4},u_{3,4})$$ Note that $[u_{1,3},u_{1,2}]=u_{2,3}$ and thus $[V,W]$ is not a Lie algebra, since $u_{2,3}\notin[V,W]$.

Well I am not a big fan of my own example so if anyone had anything more simple or more elegant, I would be interested too.

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    Thanks a lot, I don't think there is any elegant way to prove it... :)2012-11-27