database - Is it advisable to separate prime attribute partial dependencies when normalising?

Question

Suppose we have a relation R(A, B, C, D). If R has the following functional dependencies:

A, C -> D
B, C -> D
A -> B
B -> A

We should have the following candidate keys:

{A, C}, {B, C}

Thus, A, B and C are prime attributes.

I chose {A, C} as the primary key.

At this point I have arrived at an impasse.

While, from my experience, when normalising to the 2nd Normal Form it would be advisable to split the relation into:

R1(A, C, D)
R2(A, B)

But technically, following the requisites of the 2nd Normal Form, this isn't necessary as R is already in 2nd Normal Form.

Should I follow my gut and separate the relation or should I leave it as it is?

标签： databasedatabase-normalization