In this paper, the authors study the Lie algebra \(HH^1(A)\), where \(A\) is a split finite dimensional associative unital algebra over a field \(k\) and prove some interesting results. Firstly, it is shown that if the Ext-quiver of \(A\) is a simple directed graph, then the derived Lie subalgebra of \(HH^1(A)\) is nilpotent and in particular, the Lie algebra \(HH^1(A)\) is solvable. The next main result is, 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 \((k)\neq 2\) and \(HH^1(A)\cong sl_2(k)\). Finally, for an algebraically closed field \(k\), they study a finite-dimensional symmetric \(k\)-algebra \(A\) with a quiver that has a vertex with a single loop and prove that if \(HH^1(A)\) is a simple Lie algebra, then char \((k) =p >2\) and \(HH^1(A)\) is isomorphic to either \(sl_2(k)\) or the Witt Lie algebra \(W= \mathrm{Der}(k[x]/(x^p))\). They conclude the paper with few nice examples.