  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. It is
shown 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 $\mathfrak{sl}_2(\mathbb{F})$. We study the representation theory of
Hom-Lie algebras within the appropriate 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. We
apply this result to characterize the non-solvable Lie algebras having an
invertible generalized derivation of the type of $D$. Finally, using root space
decomposition techniques we provide an intrinsic proof of the fact that
$\mathfrak{sl}_2(\mathbb{F})$ is the only simple Lie algebra admitting
non-trivial Hom-Lie structures.
