I need to know how we can prove the following corollary : If $x_1, \ldots, x_n$ is a vector space basis for Lie algebra $L$ then a vector space basis for $U(L)$, $U(L)$ is universal enveloping algebra, is given by all monomials of the form $x_{j1}, x_{j2}, \ldots, x_{jl}$. I want to know how we can investigate the linearly independent of basis like those?
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I suggested an edit to improve the math, but probably my edit was not correct. The "monomial of the form ..." should be probably different (multiplication, exponentiation..?). Please, user 146566, edit your question to fix the formulation. – Peter Franek Jul 10 '14 at 09:25
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What statement of PBW are you using? And do you mean to include conditions on the $j_i$? – Aaron Jul 10 '14 at 15:47