Iwasawa decomposition of $SL(2,R)$. The order of KAN/ANK/NAK..

Question

I would like to decompose a matrix with Det=1 using the Iwasawa decomposition method. But I am confused with the proper order used in the decomposition. In Wikipedia and in CONRAD, K the Iwasawa decomposition for $SL(2,R)$ is given as $$SL2(\mathbb{R}) = KAN$$ However, in the book of V. Kisil it reads $$SL 2 (\mathbb{R}) = ANK $$ In the lectures on $SL_{2}(\mathbb{R})$ I see that $$SL_{2}(\mathbb{R})=NAK$$ Is there any difference in the above examples? I am interested in this application to matrices multiplication so the order should play a role. Thanks.

Answer 1

The $NAK$ and $KAN$ decompositions are equivalent, since the inverse map $g \mapsto g^{-1}$ reverses the order. Since $AN=NA$, we also have $NAK=ANK$.