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

We show that the number \alpha =(1+\sqrt {3+2\sqrt {5}})/2 with minimal polynomial x^4-2x^3+x-1 is the only Pisot number whose four distinct conjugates \alpha _1,\alpha _2,\alpha _3,\alpha _4 satisfy the additive relation \alpha _1+\alpha _2=\alpha _3+\alpha _4. 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 \alpha _1 = \alpha _2 + \alpha _3+\alpha _4 or \alpha _1 + \alpha _2 + \alpha _3 + \alpha _4 =0 cannot be solved in conjugates of a Pisot number \alpha . We also show that the roots of the Siegel's polynomial x^3-x-1 are the only solutions to the three term equation \alpha _1+\alpha _2+\alpha _3=0 in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation \alpha _1=\alpha _2+\alpha _3.