Where does “Gaoidhealg” come from?

Gaoidhealg (Irish) comes from Old Irish Goídelc, from Old Irish Goídel, from Old Welsh Guoidel, from Proto-Brythonic guɨðel, from Proto-Celtic wēdus, from Proto-Indo-European widʰu- — forest, wood; tree, beam.

Ancestry of “Gaoidhealg”, step by step

Step	Language	Word	Meaning
1	Old Irish	Goídelc	the Old Irish language
2	Old Irish	Goídel	Gael
3	Old Welsh	Guoidel	wild man, warrior
4	Proto-Brythonic	guɨðel	savage, woodsman
5	Proto-Celtic	wēdus	wild
6	Proto-Indo-European	widʰu-	forest, wood; tree, beam

Every word from Proto-Indo-European widʰu- →Every word from Proto-Celtic wēdus →Every word from Proto-Brythonic guɨðel →