I never thought I'd be able to make a connect Fano plane out of chemical elements, since there are so few to choose from. And I was right, but I got really close.
So if you haven't seen previous connect Fano planes, they are very much like a connect wall, except you are given seven words (element names in this case). Your job is to determine seven categories such that the elements and the categories form a Fano plane: each category matches exactly three elements, each element is in exactly three categories, each pair of distinct categories have one element in common, and each pair of elements lie together in one category. Ideally, if you get the categories correct, there will be a final answer that matches all seven categories, but I was only able to get six categories to work...so one of the categories will NOT match the final answer.
Obligatory Fano plane picture:
So here are the elements:
COPPER, GALLIUM, LUTETIUM, POLONIUM, PROTACTINIUM, RUTHENIUM, THALLIUM
Solver Note: I used everything I could think of to come up with categories. Element names, atomic numbers, atomic symbols, physical properties (I cap at high school chemistry, so don't pull out your CRC Handbook), and the periodic table are all fair game.
The answers as of 1300 UTC, 17 July 2020, contain between them four of the lines I used. A line which has not appeared yet is defined by a physical property of the elements.
To find another line that has not appeared yet, look at the periodic table.