Mathematica not recognizing expressions are real

Question

I see no problem with following calculation.

Refine[ComplexExpand[b I /(a + b I)], {Element[a, Reals], Element[b, Reals]}]

(I a b)/(a^2 + b^2) + b^2/(a^2 + b^2)

The real and imaginary parts are nicely separated.

However, the next calculations are confusing to me. Please explain how I get the output. Should I get

b^2/(a^2 + b^2) 

as I declared a and b are real?

Refine[
  Re[(I a b)/(a^2 + b^2) + b^2/(a^2 + b^2)], {Element[a, Reals], Element[b, Reals]}]

-a b Im[1/(a^2 + b^2)] + b^2 Re[1/(a^2 + b^2)]

Answer 1

You must also tell Mathematica that a^2 + b^2 is not zero.

Refine[
  Re[(I a b)/(a^2 + b^2) + b^2/(a^2 + b^2)], {a, b} ∈ Reals && a^2 + b^2 > 0]

b^2/(a^2 + b^2)