'MySQL: Extract all Keys from a nested JSON String

In addition to this answer is it possible to extract nested keys in a simple way? Example:

{
  "a": value,
  "b": {
    "c": value
    "d": {
      "e": value
   } 
  }
}

Expected output: ['a', 'b.c', 'b.d.e'] What I have tried:

 SELECT 
 f.`id` AS `field_name`
 
 FROM table t,
    JSON_TABLE(
         JSON_KEYS(t.`column`, '$.b'),
         '$[*]' COLUMNS(
           `id` VARCHAR(191) PATH '$'
         )
     ) AS t

but that would only show me one of the nested keys and skip the outer



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