Dear Christophe

You wrote:
So jy > jj can be diachronically explained by the " 'MIA' assimilation of the cluster -jy-" according to Oberlies; nevertheless, other historical explanations are possible  . . . 
 
Another historical explanation is given by this very old source: Brugmann: Elements of the Comparative Grammar of the Indo-Germanic Languages (translation by Wright) .  He suggests sajjate is from Indo-Iranian *sa-zj-a-
(pages 448-449)  Note:his abbreviation Ar. = Indo-Iranian.

Prim. Ar. azg = Skt. adg Av. azg,. . . But Prim.  Ar. azj passed through adj into ajj in Skr. (cp. ujjiti-ṣ 'victory from *ud-jiti-ṣ section 355). majjāmi 'I duck' majjan- 'marrow' from the same root mezg-. sajjate 'clings to something' from *sa-zj-a- redupl. (cp. sa-sc-a-ti 'follows' from sac'): cp. Lith. segu 'I fasten'.

Harry Spier