Why does the keyword metadata not display in Acrobat Reader, but does in other readers?

Worth noting that the Info dictionary, and hence /Keywords, is obsolete in PDF 2.0. Conforming PDF 2.0 readers ignore it, I think. For many years Acrobat has set both Info and XMP metadata. 

(Source: answer to the original question).

In this PDF file, metadata are stored both in old and XMP format.

Using cpdf command as shown below:

cpdf synthèse_etude_canaux_communication.pdf -info

... Would show the metadatas stored in these formats:

• Title: <title>
• Author: <author>
• Keywords: <keywords>
• Creator: Microsoft PowerPoint 2019
• Producer: Microsoft PowerPoint 2019
• ...
• Created: D:20210713102823+02'00'
• Modified: D:20210921101251+02'00'
• XMP pdf:Producer: Microsoft® PowerPoint® 2019
• XMP xmp:CreateDate: 2021-07-13T10:28:23+02:00
• XMP xmp:CreatorTool: Microsoft® PowerPoint® 2019
• XMP xmp:MetadataDate: 2021-09-21T10:12:51+02:00
• XMP xmp:ModifyDate: 2021-09-21T10:12:51+02:00
• XMP dc:title: <title>
• XMP dc:creator: <author>

Not yet confirmed by a source, but it seems that Adobe Acrobat prefer XMP over old-style metadata, not mixing them, and that's why the keywords aren't displayed in the document properties.