The purpose of this paper is to show that there are Hom-Lie algebra
structures on $\mathfrak{sl}_2(\mathbb{F}) \oplus \mathbb{F}D$, where $D$ is a
special type of generalized derivation of $\mathfrak{sl}_2(\mathbb{F})$, and
$\mathbb{F}$ is an algebraically closed field of characteristic zero. We study
the representation theory of Hom-Lie algebras within the approriate category
and prove that any finite dimensional representation of a Hom-Lie algebra of
the form $\mathfrak{sl}_2(\mathbb{F}) \oplus \mathbb{F}D$, is completely
reducible, in analogy to the well known Theorem of Weyl from the classical Lie
theory. It turns out that the generalized derivations $D$ of
$\mathfrak{sl}_2(\mathbb{F})$ that we study in this work, satisfy the Hom-Lie
Jacobi identity for the Lie bracket of $\mathfrak{sl}_2(\mathbb{F})$. It is a
known result that $\mathfrak{sl}_2(\mathbb{F})$ is the only simple Lie algebra
admitting non-trivial Hom-Lie structures. An intrinsic proof of this fact using
root space decomposition techniques is given.