'Lorrain' (no e) is generally accepted to be a Daum mark. However, it does get a little confusing when you look further. Ivo mentions 'Lorrain' as the 'Trade name used by Daum for art glass produced at verrerie de Croismare' (in Millers Glass fact file a-z). However he also lists Verrerie d'art de Lorrain in a separate entry as a glassworks founded by Paul Daum to produce cheap cameo glass...
The name Verreries de Belle-Etoille is also mentioned with dates of c1927-30. Apparently this is the works where D'Avesn was creative director and which produced the 'Lorrain' pieces. Not sure whether this was the same works as Verreries d'Art de Lorrain or not though but it seems plausible...
To add to the confusion there's also a Verreries d'Art de Lorraine (with 'e') listed as a Meisenthal trademark. Regarding the mark 'VAL,' it's often said to refer to Daum but Im not sure if theres any evidence to support this. Cappa also mentions a 'Val et Cie.' as a separate maker.
So, basically, the more I read the more confused I get...
