For the given relation R (A,B,C,D,E,F,G,H) we have the following candidate keys: AD, BD, DE, DF.
Step 1:
A--> BC is a partial dependency.
Let us decompose R(A,B,C,D,E,F,G,H) to R1(A,B,C,E,F,G,H) and R2(A,D)
Union of R1 and R2 has all the attributes of R and the intersection A is the primary key of R1
Till now all functional dependencies are preserved.
R2(A,D) is in BCNF
In R1, A,B, E and F are candidate keys, However CH is not a super key. So we decompose further.
Step 2:
R1(A,B,C,E,F,G,H) decomposes into R3(A,B,C,E,F,H) and R4(C,H,G)
Union of R3 and R4 has all attributes of R2 and the intersection CH is primary key of R4
Dependencies in R4: CH-->G
CH is super key of R4, so R4 is in BCNF
Dependencies in R3: A-->BC, B-->CFH , E-->A, F-->EG
A,B,E,F are super keys in R3 so R3 is in BCNF
R (A,B,C,D,E,F,G,H) can be thus decomposed into R'(A,D) , R''(A,B,C,E,F,H) and R'''(C,H,G)