simplifying expressions - Why doesn't Mathematica make an obvious simplification?

I am doing some calculation with summation and the Kronecker symbol. Here are my steps :

$Assumptions = 
  k1 ∈ Reals &&  k2 ∈ Reals &&  k3 ∈ Reals &&  p1 ∈ Reals &&  p2 ∈ Reals &&  p3 ∈ Reals
k = {k1, k2, k3};
p = {p1, p2, p3};
d[i_, j_] := KroneckerDelta[i, j]
proj[i_, j_, k1_, k2_, k3_] := 
  d[i, j] - 
   (d[i, 1]*k1 + d[i, 2]*k2 + d[i, 3]*k3)*

   (d[j, 1]*k1 + d[j, 2]*k2 + d[j, 3]*k3)/
   (k1^2 + k2^2 + k3^2)
test1 = proj[i, j, k1, k2, k3]*proj[i, j, p1, p2, p3];
test2 = Sum[Sum[test1, {i, 1, 3}], {j, 1, 3}]
test2 // Expand

To explain the steps:

1) I define $\vec{k}$ and $\vec{p}$ with real components.
2) I define a projector $P_{ij} \left( \vec{k} \right) = \delta_{ij} - \frac{k_i k_j}{k^2}$.
3) I compute a summation on the repeated subscript.

After the last step, I have a relatively big expression, the product of k and p components. It looks like

$$3-\frac{a}{a+b+c} - \frac{b}{a+b+c} - \frac{c}{a+b+c} +...-...$$ The a, b and c stands for k1, k2 and k3 (or p1, 2, 3).

Now the question: why doesn't Mathematica make the simplification because, as anyone can see, the preceding expression can be simplified to $2 +...-...$

Is the problem linked to the Expand operation? How can I make the simplification I want?. I thought of using /. to do it, but that doesn't work either.

I hope someone will understand my question!

Answer

You have to explicitly tell Mathematica to simplify expressions. You can do this using Simplify or FullSimplify

Simplify@test2

(2 k2 k3 p2 p3 + 2 k1 p1 (k2 p2 + k3 p3) + 
   k1^2 (2 p1^2 + p2^2 + p3^2) + k2^2 (p1^2 + 2 p2^2 + p3^2) + 

   k3^2 (p1^2 + p2^2 + 2 p3^2))/((k1^2 + k2^2 + k3^2) (p1^2 + p2^2 + 
     p3^2))