Complex Conjugation Confusion

Hi

I seem to have contradicting rules on how to treat complex conjugates

where we have

a = x + iy

a*= x - iy

but in an example showing how you can find whether a complex expression is real or imaginery we have

b = i(cd* - c*d)

b* = -i(c*d - cd*)

so to compare to my previous expression

if we say x = 0 and y = (cd* - c*d)

we have the first example saying

a = x + iy

a*= x - iy

and the second saying

b = x + iy

b*= x - iy*

is it simply that the * is dropped when a variable is known to be real ?

I am new to this subject and any insights would be most welcome !