  We show that there exists a Banach space $E$ with the following properties:
the Banach algebra $\mathscr{B}(E)$ of bounded, linear operators on $E$ has a
singular extension which splits algebraically, but it does not split strongly,
and the homological bidimension of $\mathscr{B}(E)$ is at least two. The first
of these conclusions solves a natural problem left open by Bade, Dales, and
Lykova (Mem. Amer. Math. Soc. 1999), while the second answers a question of
Helemskii. The Banach space $E$ that we use was originally introduced by Read
(J. London Math. Soc. 1989).
