In a previous question (The Iwasawa decomposition of $\text{GL}(2,\mathbf R)$), it was noted that the Iwasawa decomposition of GL(2,R) is:
$$ \underbrace{\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}}_K \underbrace{ \begin{bmatrix} r_1 & 0 \\ 0 & r_2 \end{bmatrix}}_A \underbrace{\begin{bmatrix}1&x\\ 0 & 1 \end{bmatrix}}_N $$
What is the lie Algebra of each term? Can I write the decomposition as $\exp B \exp C \exp D$, where $K=\exp B$, $A=\exp C$ and $N=\exp D$. What are B, C and D?
