  Let $A$ be a split finite-dimensional associative unital algebra over a
field. The first main result of this note shows that if the Ext-quiver of $A$
is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The
second main result shows that if the Ext-quiver of $A$ has no loops and at most
two parallel arrows in any direction, and if $HH^1(A)$ is a simple Lie algebra,
then char(k) is not equal to $2$ and $HH^1(A)\cong$ $sl_2(k)$. The third result
investigates symmetric algebras with a quiver which has a vertex with a single
loop.
