No two non-real conjugates of a Pisot number have the same imaginary parts

We show that the number α=(1+3+25√−−−−−−−√)/2 with minimal polynomial x4−2x3+x−1 is the only Pisot number whose four distinct conjugates α1,α2,α3,α4 satisfy the additive relation α1+α2=α3+α4. This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the other hand, we prove that similar four term equations α1=α2+α3+α4 or α1+α2+α3+α4=0 cannot be solved in conjugates of a Pisot number α. We also show that the roots of the Siegel's polynomial x3−x−1 are the only solutions to the three term equation α1+α2+α3=0 in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation α1=α2+α3.