  We construct a new representation of the infinite rank Lie algebra
$a_{\infty}$ with central charge $c=1$ on the Fock space $\mathit{F^{\otimes
\frac{1}{2}}}$ of a single neutral fermion. We show that $\mathit{F^{\otimes
\frac{1}{2}}}$ is a direct sum of irreducible integrable highest weight modules
for $a_{\infty}$ with central charge $c=1$. We prove that as $a_{\infty}$
modules $\mathit{F^{\otimes \frac{1}{2}}}$ is isomorphic to the Fock space
$\mathit{F^{\otimes 1}}$ of the charged free fermions. As a corollary we obtain
the decompositions of certain irreducible highest weight modules for
$d_{\infty}$ with central charge $c=\frac{1}{2}$ into irreducible highest
weight modules for $d_{\infty}$ with central charge $c=1$.
