  The main goal of this paper is to prove the following theorem:
  Let $\frak k$ be an $\frak {sl}_2$-subalgebra of a semisimple Lie algebra
$\frak g$, none of whose simple factors is of type $A1$. Then there exists a
positive integer $b(\frak k, \frak g)$, such that for every irreducible finite
dimensional $\frak g$-module $V$, there exists an injection of $\frak
k$-modules $W \to V$, where $W$ is an irreducible $\frak k$-module of dimension
less than $b(\frak k, \frak g)$.
  This result was announced in math.RT/0310140.
