The Byzantine chapel is a sort of autonomous Russian Catholic chapel, and since there is no Russian Catholic hierarch currently, is subject to the local Latin bishop who has never interferred with the community and to the contrary has greatly supported their work.
The Rule of St. Benedict is strictly a collection of statutes for monastic life and is not specific as to the liturgical life of the monastery, and thus could be used in any liturgical tradition.
Their Kyivan/Obikhod chant in Church Slavonic is excellent and even some Orthodox have studied with them. They also publish many excellent books and journals there including Irenikon which is an exceptional academic journal dealing with Eastern Christian matters. They undertook the translation of some of Lev Gillet's works. Fr. Lev Gillet, along with Metropolitan Sheptytsky, were primary inspirations for Dom Lambert's creation of the monastery of Chevtogne.
If you want a very good CD of a typical Saturday Night Vespers with Kyivan/Obikhod chant I highly recommend their "Russische Klostervesper" which starts off with a nice semadron rap. It is distributed by Koch/Schwann and believe it or not I found one at the local Barnes and Noble.
It was Dom Lambert's vision to show with two liturgical traditions living in one monastic community that monasticism is a common foundation of the universal Church with the monastery at Chevtogne, which first began at Amay.
It's interesting that the fresco in the Latin church, only recently completed in the mid 1990s, was the work of Archimandrite Zenon from Pskov.
Also, there is an address from Ecumenical Patriarch +Bartolomeos about Dom Lambert which was given at Chevtogne at the Patriarchate's website, http://www.patriarchate.org/SPEECHES/1994/Belgium_address.html
The late Father Alexander Schmemann also considered Dom Lambert and his work at Chevtogne to be an inspiration to him as well.
Dom Lambert was one of the greatest proponents of moving worship and liturgy into parishes from the monasteries, including parish celebrations of the Divine Office.