For an arbitrary finite-dimensional algebra
A
A
, we introduce a general approach to determining when its first Hochschild cohomology
H
H
1
(
A
)
\mathrm {HH}^1(A)
, considered as a Lie algebra, is solvable. If
A
A
is, moreover, of tame or finite representation type, we are able to describe
H
H
1
(
A
)
\mathrm {HH}^1(A)
as the direct sum of a solvable Lie algebra and a sum of copies of
s
l
2
\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
A
. As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll, and Solotar.