  For an arbitrary finite-dimensional algebra $A$, we introduce a general
approach to determining when its first Hochschild cohomology ${\rm HH}^1(A)$,
considered as a Lie algebra, is solvable. If $A$ is moreover of tame or finite
representation type, we are able to describe ${\rm HH}^1(A)$ as the direct sum
of a solvable Lie algebra and a sum of copies of $\mathfrak{sl}_2$. We proceed
to determine the exact number of such copies, and give an explicit formula for
this number in terms of certain chains of Kronecker subquivers of the quiver of
$A$. As a corollary, we obtain a precise answer to a question posed by
Chaparro, Schroll and Solotar.
