Every derivative $\delta : L \to L $ for finite semisimple Lie algebra $L$ over $\Bbb C $ is inner automorphism

Question

I need to show that Every derivative $\delta : L \to L $ for finite semisimple Lie algebra $L$ over $\Bbb C $ is inner automorphism.

when Inner automorphism is defined by $exp(ad_x)= \sum_{j=0}^{k-1} ad_x^j/j!$

when $ad_x$ nilpotent with nilpotent degree of $k$

for all $x \in L$ when $ad_x:L \to L , ad_x(y)=[x,y] $