Let L be a finite-dimensional complex semisimple lie algebra, then ad(L)=Der(L). (Der is short for derivation). In order to show that ad(L)=Der(L), the proof I followed proves that that the prependicular space to ad(L) is zero. (Perpendicular with respect to killing form of Der(L).) I understand the proof of this. But why is this condition sufficient to conclude ad(L) = Der(L)

If we knew that Der(L) is the direct sum of ad(L) and P:=Perp(ad(L)) then we would be done. And non-degeneracy of the killing form restricted to ad(L) tells us that it is sufficient to show Dim(Der(L)) = Dim(ad(L)) + Dim(P). But I am failing to see how we have that.