A great addition would be an icon for the periodic system of elements. Unfortunately I have no excellent idea on how to visualize this. An outline or shape of the PSE would not be easily recognizable.
Would atom not work for you?
Thanks! That's what I'm currently using. But I'm not sure if the user will recognize it as the element picker button:

How about this then? I know that it's in English, but this is the international symbol for helium, so I don't believe that we'll have an issue with that. The only issue is that it takes up the full 24-pixel height and 22-pixel width. I'm not sure how I feel about that.

Definitely a nice idea, but I think using letters to represent this is going to be too niche. Plus it doesn't align with the goal of using symbols over text. Inverting the colors would allow for better sizing.
Thinking about closing this if there is no objection.
@pijnappel What about a simple table icon (like this)?
I liked the proposal of @MrGrigri
This is my adaptation:

Element1.zip
How about an approximation of the p-table shape?
Actually that's pretty recognizable. Reopening.
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How about an approximation of the p-table shape?