Decomposing a relation into BCNF

I'm having trouble establishing when a relation is in Boyce-Codd Normal Form and how to decompose it info BCNF if it is not. Given this example:

R(A, C, B, D, E) with functional dependencies: A -> B, C -> D

How do I go about decomposing it?

The steps I've taken are:

A+ = AB  
C+ = CD  
R1 = A+ = **AB**  
R2 = ACDE (since elements of C+ still exist, continue decomposing)  
R3 = C+ = **CD**  

R4 = ACE (no FD closures reside in this relation)

So now I know that ACE will compose the whole relation, but the answer for the decomposition is: AB, CD, ACE.

I suppose I'm struggling with how to properly decompose a relation into BCNF form and how to tell when you're done. Would really appreciate anyone who can walk me through their thought process when solving these problems. Thanks!