§ Splitting of semidirect products in terms of projections

Say we have an exact sequence that splits:
0NiGπK0 0 \rightarrow N \xrightarrow{i} G \xrightarrow{\pi} K \rightarrow 0
with the section given by s:KGs: K \rightarrow G such that kK,π(s(k))=k\forall k \in K, \pi(s(k)) = k. Then we can consider the map πkspi:GG\pi_k \equiv s \circ pi: G \rightarrow G. See that this firsts projects down to KK, and then re-embeds the value in GG. The cool thing is that this is in fact idempotent (so it's a projection!) Compute:
πkπk=(sπ)(sπ)=s(πs)π)=sidπ=sπ=πk \begin{aligned} &\pi_k \circ \pi_k \\ &= (s \circ \pi) \circ (s \circ \pi ) \\ &= s \circ (\pi \circ s) \circ \pi ) \\ &= s \circ id \circ \pi \\ &= s \circ \pi = \pi_k \\ \end{aligned}
So this "projects onto the kk value". We can then extract out the NN component as πn:GG;πn(g)gπ(k)1\pi_n: G \rightarrow G; \pi_n(g) \equiv g \cdot \pi(k)^{-1}.