  In 2014, we determine the precise form of a continuous orthogonal form on a
commutative real C$^*$-algebra. We also describe the general form of a
(not-necessarily continuous) orthogonality preserving linear map between
commutative unital real C$^*$-algebras. Among the consequences, we show that
every orthogonality preserving linear bijection between commutative unital real
C$^*$-algebras is continuous. In this note we revisit these results and their
proofs with the idea of filling a gap in the arguments, and to extend the
original conclusions.
