How can I prove these two field extensions are equal?

Q(√3, -√3, i, -i) = Q(√3+i) where Q is the field of the rational numbers.

I got Q(√3+i) ⊆ Q(√3, -√3, i, -i), that direction is easy.

How can I prove Q(√3, -√3, i, -i) ⊆ Q(√3+i)?

Let m ∈ Q(√3, -√3, i, -i)

Therefore m = q + a√3 + bi + c(-√3) + d(-i) where q,a,b,c,d ∈ Q

Let p = a-c and r = b-d. Therefore, p,r ∈ Q

Therefore m = q + p√3 + ri

But I have to prove m = q + p√3 + ri = q + (SOMETHING IN Q)* (√3+i)

Any hints?