Wrestling with substitution rules

Question

Summary

I am struggling with substitution rules.

Example

Here are several cases which are problematic:

Clear[a,α];
{a + 2 b + 1, -a - b, 2 a + 2 b, a^2 + 2 a b + b^2} /. {a + b -> α}

Actual result:

{1 + a + 2 b, -a - b, 2 a + 2 b, a^2 + 2 a b + b^2}

Desired result:

{1 + α +  b, -α, 2 α, α^2}

Question

Currently, the rule is permuted for every case, e.g.

2 a + 2 b -> 2α

Can the alpha substitution rule be generalized That, is there a single rule to handle all cases?

Answer 1

{a + 2 b + 1, -a - b, 2 a + 2 b, a^2 + 2 a b + b^2} /. {a -> α - b} // Simplify

{1 + b + α, -α, 2 α, α^2}

Answer 2

Also

Simplify[{a + 2 b + 1, -a - b, 2 a + 2 b,  a^2 + 2 a b + b^2}, {a + b == α}]

{1 + b + α, -α, 2 α, α^2}