Let \(K\) be a field of characteristic 0 and \(A\) a \(K\)-vector space. A left symmetric algebra structure on \(A\) is a bilinear product \((x,y) \mapsto x\cdot y\) satisfying \[ x\cdot (y\cdot z)- (x\cdot y) \cdot z=y \cdot(x \cdot z)- (y\cdot x) \cdot z. \tag{*} \] The condition (*) implies that the commutator \([x,y]:=x\cdot y-y\cdot x\) turns \(A\) into a Lie algebra which we denote by \({\mathfrak g}\). If \((A,\cdot)\) is a finite dimensional left symmetric algebra and \(R(a):b \mapsto b\cdot a\) is the right regular representation of \({\mathfrak g}\) on \(A\), then \(A\) is called complete if \(R(a)\) is traceless for all \(a\in A\). In this paper the author determines all simple left symmetric algebras of dimensions 2 and 3 as well as the complete simple left symmetric algebras of dimension 4. The tools involved are deformation theory and some structure theory arising from a comparison of the left regular representation of \({\mathfrak g}\) on \(A\) with the adjoint representation of \({\mathfrak g}\).