serodiagnostically

The story

1. Proto-Indo-European*ser-
2. Proto-Indo-European*sér-o-
3. Proto-Italic*serom
4. Latinserum
5. Englishserum
6. Englishsero-
7. Proto-Indo-European*dwóh₁
8. Proto-Indo-European*trísinflu.
9. Proto-Indo-European*dwís
10. Ancient Greekδιά (diá)
11. Ancient Greekδια- (dia-)
12. Proto-Indo-European*ǵneh₃-
13. Proto-Indo-European*ǵn̥h₃sḱéti
14. Proto-Hellenic*gignṓskō
15. Ancient Greekγῐγνώσκω (gĭgnṓskō)
16. Ancient Greekδῐαγῐγνώσκω (dĭagĭgnṓskō)
17. Ancient Greekδῐᾰ́γνωσῐς (dĭắgnōsĭs)
18. Ancient Greekδῐᾰ- (dĭă-)
19. Proto-Indo-European*ǵneh₃-
20. Proto-Indo-European*ǵneh₃tis
21. Ancient Greekγνῶσις (gnôsis)
22. Ancient Greekγνωστῐκός (gnōstĭkós)
23. Ancient Greekδῐαγνωστῐκός (dĭagnōstĭkós)
24. Englishdiagnostic
25. Englishserodiagnostic
26. Proto-Indo-European*h₂el-
27. Proto-Indo-European*h₂élisder.?
28. Proto-Indo-European*leyg-
29. Proto-Germanic*līką
30. English todayserodiagnostically

From serodiagnostic + -ally. By surface analysis, sero- + diagnostically.