More complicated.
Laffiteau bought the Lefaucheux name from the wife of Casimir (for 150,000FF!).
Rieger bought it from Laffiteau.
But Eugene Lefaucheux opened shop under his own name at the same time.
So there was a Lefaucheux brand (37 rue Vivienne) (owned by Laffiteau and then Rieger) and a E. Lefaucheux brand ( Rue Lafayette) at the same time.

I don't know when Rieger folded or even when he died, but both houses were bought by Mode eventually ( as well as Pirlet) at least by 1919.

I wish there were a book about Parisian outfits at the 19th/20th centuries junction. That would be quite helpful.
However, one would need to live there to research this.

All the best,

WC-