# fa.functional analysis – Density of a graded algebra

I’m trying to prove the following proposition:

## If $$v in V$$ and $$Y in mathfrak{so} (V)$$ then $$(dotmu(Y), B(v)) = B(Yv)$$.

By definition
$$(dotmu(Y), B(v)) = dotmu(Y)B(v) – B(v)dotmu(Y).$$ I arrive at
$$dotmu(Y),B(v),f_S = B(v),dotmu(Y),f_S + B(Yv),f_S , iff B(Yv), f_S = dotmu(Y),B(v),f_S – B(v),dotmu(Y),f_S,.$$

The set $${f_S: g cdot S : : mathrm{exists}}$$
generates a dense subspace of $$mathcal F ^ + (V)$$, the even subalgebra of $$mathcal F (V)$$. But this doesn’t imply that $${f_S: g cdot S : : mathrm{exists}}$$ is dense in $$mathcal F (V)$$ since density is not commutative. ¿Any idea on how to conclude this?

Note: Here $$mathcal F (V)$$ is the Fermionic Fock space: $$bigoplus_{k=0}^m V^{wedge k}$$, where $$V$$ is a (complex) vector space and $$mathfrak{so} (V)$$ is the Lie algebra of the special ortogonal group.

Thank you :).