An Elementary Connect Fano Plane

Question

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.

Good luck!

HINT #1:

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.

HINT #2:

To find another line that has not appeared yet, look at the periodic table.

Answer 1

After a LOT of false trails and dead ends I believe I have FINALLY hit upon the solution!

PROTACTINIUM, LUTETIUM, GALLIUM - atomic numbers comprise only odd numbers (#91, #71, and #31, respectively);

PROTACTINIUM, COPPER, POLONIUM - sum of digits in atomic number is greater than or equal to 10 (#91 = 10, #29 = 11, and ##84 = 12, respectively);

PROTACTINIUM, THALLIUM, RUTHENIUM - density greater than 10 g/cm³ (11.72, 11.85, and 12.37, respectively);

LUTETIUM, COPPER, RUTHENIUM - second letter of symbol is 'u' (Lu, Cu, and Ru, respectively);

GALLIUM, COPPER, THALLIUM - diagonally adjacent to Cadmium (Cd) in the Periodic Table;

GALLIUM, POLONIUM, RUTHENIUM - named after countries on the European mainland (France, Poland, and Russia, respectively);

LUTETIUM, THALLIUM, POLONIUM - occupy the 6th row (period 6) of the Periodic Table.

There is an element which meets 6 of these 7 criteria, as set out by the OP, which is:

GOLD

- Atomic number is 79 - both odd digits;
- Digits of atomic number sum is greater than 10 (7+9=16);
- Density is greater than 10 g/cm³ (19.32);
- Second letter of symbol is 'u' (Au);
- Diagonally adjacent to Cadmium in the Periodic Table;
- Is not named after a country on the European mainland;
- Occupies the 6th row of the Periodic Table.

And for ease of checking some of the Periodic Table-based criteria, here's a copy of it with the relevant elements highlighted:

Answer 2

Edit: Not the right answer but may serve as a partial as OP has suggested some of the categories match the intended solution (I misread the question before, there is a final answer which should belong to six of the constructed categories).

Configuration

                 POLONIUM


        GALLIUM          PROTACTINIUM

                 THALLIUM

 COPPER          LUTETIUM               RUTHENIUM

Groups

{COPPER, LUTETIUM, RUTHENIUM} - chemical symbols contain letter U.
{COPPER, THALLIUM, PROTACTINIUM} - chemical symbols don't match first letters of name.
{COPPER, GALLIUM, POLONIUM} - named after EU countries (Cyprus, France, Poland).
{POLONIUM, PROTACTINIUM, RUTHENIUM} - all end in NIUM.
{POLONIUM, THALLIUM, LUTETIUM} - all in period 6.
{GALLIUM, PROTACTINIUM, LUTETIUM} - atomic numbers contain two odd digits.
{GALLIUM, THALLIUM, RUTHENIUM} - all belong to groups which are Fibonacci numbers.

Answer 3

I've been able to come up with a number of categories. Here are three of them as a partial answer (ASCII art because I'm having trouble figuring out the current state of LaTeX tables on SE):

$ \begingroup \small \newcommand \row[8] {#1&\sf #2&\sf #3&\sf #4&\sf #5&\sf #6&\sf #7&\sf #8} \begin{array}{l|ccccccc} \row{ }{Cu}{Ga}{Ru}{Lu}{Tl}{Po}{Pa}\\ \hline \row{\textsf{named after a country}^*}{ }{ X}{ X}{ }{ }{ X}{ }\\ \row{\textsf{prime atomic number} }{ X}{ X}{ }{ X}{ }{ }{ }\\ \row{\textsf{post-transition metal} }{ }{ X}{ }{ }{ X}{ X}{ }\\ \end{array} \endgroup $

* a recognized country at the time of the element's naming