Can we decompose a schema into BCNF when it is not even in 1NF
Always hit it at first shot? It is not guaranteed. You apply normalization rules to remove anomalies from relations. Relations rise to a more normal form. You observe for anomalies again and if there are anomalies apply rules again. It is possible to reach most normal form possible for the relation at first move (as you know there is a limit for normalization of any relation. Ron Fagin showed that it is often impossible to achieve DKNF form) but not always. Ron Fagin's paper may help.