Set Up:
Let $ K $ be a totally real number field. Let $ \mathcal{O}_K $ be the ring of integers of $ K $. Let $ G $ be a simple linear algebraic group. Suppose that $ G(\mathbb{R}) $ is a compact Lie group. $ G(\mathcal{O}_K) $ must be discrete since it is an arithmetic group and then $ G(\mathcal{O}_K) $ is a discrete subgroup of the compact group $ G(\mathbb{R}) $ so it must be finite.
Question:
What can we learn about the finite group $ G(\mathcal{O}_K) $? For example, using the assumption that $ G $ is simple, can we prove that $ G(\mathcal{O}_K) $ is 2-generated?
Motivation:
Finite simple groups are known to be 2-generated Generating finite simple groups with $2$ elements , maybe something similar carries over here?
Another slightly similar fact is that compact simple Lie groups are topologically 2-generated and moreover the two generators can be chosen to have finite order Semisimple compact Lie group topologically generated by two finite order elements
Cross-posted from MSE https://math.stackexchange.com/questions/4742994/are-the-integer-points-of-a-simple-linear-algebraic-group-2-generated
