RMQ to LCA Reduction

I showed LCA reduces to RMQ (via Euler tour). The reverse is also true: RMQ reduces to LCA.

Given an array, build a Cartesian tree where each node's value is the minimum of its subtree.

Then $\text{RMQ}(l, r)$ equals the value at $\text{LCA}(\text{node}_l, \text{node}_r)$. Why care?

Some LCA structures (like binary lifting) are simpler to implement than sparse tables. Also, this connection is theoretically interesting: RMQ and LCA are equivalent problems. Any $O(n)$ preprocessing, $O(1)$ query solution for one gives the same for the other.