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Is there a simple criterion to determine whether, given a set of generators $A_1,\ldots,A_n$ of a Lie subalgebra $\mathfrak{h}\subset\mathfrak{g}$, adding a new element $B\in\mathfrak{g}$ will enlarge the subalgebra? Clearly if $B$ is in the linear span of $A_1,\ldots,A_n$, the subalgebra is not enlarged, but if this is not the case, is there anything to do other than actually list out all of the nested commutators of the $A_i$ and check that $B$ is linearly independent of them?

Jeffrey
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    Generators and relations for a Lie algebra are defined by a quotient $L/R$, where $L$ is free on a set $X$, and relator ideal $R$, see here, $9.4$. It seems that you understand something different by "generator". A "new " linear independent vector $B$ need not be a new generator. – Dietrich Burde Jan 28 '16 at 09:51
  • Yes, I don't mean generator in the sense of free algebras, but in the sense that $A_1,\ldots,A_k$ are a set of generators for a Lie algebra $\mathfrak{g}$ if $\mathfrak{g}$ is the smallest Lie algebra containing the $A_i$. Is there another name? An (obviously trivial) example of what I'm asking is, suppose $\mathfrak{h}$ is the smallest Lie subalgebra of $\mathfrak{su}(2)$ containing $\left\lbrace X,Y\right\rbrace\subset\mathfrak{su}(2)$ the usual Pauli matrices and $\mathfrak{h}'$ is the smallest Lie algebra containing $\left\lbrace X,Y,Z\right\rbrace$, is $\mathfrak{h}=\mathfrak{h}'$? – Jeffrey Jan 28 '16 at 17:43

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