  Fake projective planes are smooth complex surfaces of general type with Betti
numbers equal to that of the usual projective plane. Recent explicit
constructions of fake projective planes embed them via their bicanonical
embedding in $\mathbb P^9$. In this paper, we study Keum's fake projective
plane $(a=7, p=2, \{7\}, D_3 2_7)$ and use the equations of \cite{Borisov} to
construct an embedding of fake projective plane in $\mathbb P^5$. We also
simplify the 84 cubic equations defining the fake projective plane in $\mathbb
P^9$.
