  $\newcommand{\OO}[1]{\mathcal{O}_{#1}}\newcommand{\GG}{\mathcal{G}}\newcommand{\End}{\mathrm{End}}\newcommand{\O}{\mathcal{O}}$Let
$K/F$ be a quadratic extension of non-Archimedean local fields of
characteristic not equal to 2, with rings of integers denoted by $\OO K$ and
$\OO F$. We consider a formal $\OO F$-module $\GG$, over a discrete valuation
ring $\OO W$ with an uniformizer $\varpi$, with extra endomorphisms by a
subring $\O$ of $\OO K$, and the height of its reduction $\GG_0=\GG\otimes \OO
W/\varpi$ is 2. The endomorphism ring of $\GG_n=\GG\otimes \OO W/\varpi^{n+1}$
is a subring between $\OO s$ and $\OO D=\End(\GG_0)$. We will determine them
explicitly. This result was previously proved by Gross and Keating. Their
treatment is the formal cohomology theory. We will give another proof using the
intersection formula of CM cycles in Lubin-Tate deformation spaces.
