PRIMARY index not used if too few entries
It seems you can force MySQL to use an index with force index, documented here. 

First, make sure your statistics are up to date.

Then try something like this (untested)

SELECT cmd.cmd_date, liv.liv_nom
FROM cmd 
    LEFT JOIN liv force index (primary) ON cmd.liv_id = liv.liv_id
WHERE 1
ORDER BY cmd_date DESC


Different index hints might work better. Expect different behavior when liv contains only 5 or 6 rows total rather than 5 or 6 matching rows out of thousands or millions of rows.