  Let $G$ be a connected reductive algebraic group defined over an
algebraically closed field $k$. The aim of this paper is to present a method to
find triples $(G,M,H)$ with the following three properties. Property 1: $G$ is
simple and $k$ has characteristic 2. Property 2: $H$ and $M$ are closed
reductive subgroups of $G$ such that $H<M<G$, and $(G,M)$ is a reductive pair.
Property 3: $H$ is $G$-completely reducible, but not $M$-completely reducible.
We exhibit our method by presenting a new example of such a triple in $G=E_7$.
Then we consider a rationality problem and a problem concerning conjugacy
classes as important applications of our construction.
