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When asked to replace P1^2+P2^2+P3^2 by P, GiNaC (symbolics back-end for Sage) looks for an exact match. If this sum is part of an expression, such as P1^2 + P2^2 + P3^2 + sqrt(P)*P0 there will be no substitution.

Try this instead:

sage: denominator.substitute({P1^2: P- P2^2 - P3^2}) 
-1/2*((P3 - sqrt(P))*(P1*m/(sqrt(P)*P0 + P) - I*P2*m/(sqrt(P)*P0 + P) + 2*P1 - 2*I*P2) - (P1 - I*P2)*(P3*m/(sqrt(P)*P0 + P) + 2*P0 + 2*P3 - sqrt(P)*m/(sqrt(P)*P0 + P)))/sqrt(1/2*P3*m/(sqrt(P)*P0 + P) + P0 + P3 - 1/2*sqrt(P)*m/(sqrt(P)*P0 + P))

More information available in the relevant section of the GiNaC tutorial.