A relation with only two attributes is always in BCNF.

https://stackoverflow.com/questions/33455459/how-is-every-binary-relation-bcnf

For any relation to be in BCNF, the following must holds for any functional dependency $X \rightarrow Y :$

Either $X \rightarrow Y $ is a Trivial Functional Dependency (i.e. $Y \subseteq X $) OR $X$ is a superkey for schema $R.$

A relation with only two attributes is always in BCNF.

For Example, Assume relation $R = \{A,B \}$ with two attributes. The only possible Non-trivial FD's are $ \{A \} \rightarrow \{B \}$ and $ \{B \} \rightarrow \{A \}.$

So, there are four possible cases:

1. No Non-trivial FD's hold in $R.$ i.e. $ \{Cand.Key = AB \},$ Since it is an all key relation it's always in $BCNF.$

2. Only Non-trivial FD $A \rightarrow B$ holds. In this case, $ \{Cand.Key = A \},$ and relation satisfies BCNF.

3. Only Non-trivial FD $B \rightarrow A$ holds. In this case, $ \{Cand.Key = B \},$ and relation satisfies BCNF.

4. Both $A \rightarrow B$ and $B \rightarrow A$ holds. In this case, $ \{Cand.Keys = A,B \},$ and

relation satisfies BCNF.

Hence, every Binary Relation (A relation with two attributes) is always in BCNF!