Problem #1 :

Clear["Global`*"]

g=DiagonalMatrix[{1,-1,-1,-1}];
g //MatrixForm

p1={e1,0,0,+pz};
p2={e2,0,0,-pz};
p12=p1+p2;

p2.g.p2

p12.g.p12

eqs={p1.g.p1==m12,p2.g.p2==m22,p12.g.p12==s};
eqs //TableForm

del[a_,b_,c_]=(a^2)+(b^2)+(c^2)-2(a b + b c + c a);
del[a,b,c]

Problem #2 :

a) CMS System

Clear["Global`*"]

g=DiagonalMatrix[{1,-1,-1,-1}];
g //MatrixForm

(* BEFORE *)
p1={e1,0,0,+pz};
p2={e2,0,0,-pz};
p12=p1+p2;
e1=e2=e; (* set energies equal *)

sBefore= p12.g.p12

(* AFTER *)
p3={m,0,0,0};
p4={m,0,0,0};
p5={m,0,0,0};
p6={m,0,0,0};
pAFTER=(p3+p4+p5+p6)
e1=e2=e; (* set energies equal *)

a) Fixed Target System

Clear["Global`*"]

g=DiagonalMatrix[{1,-1,-1,-1}];
g //MatrixForm

(* BEFORE *)
p1={e,0,0,+pz};
p2={m,0,0,0};
p12=p1+p2;

sBefore= p12.g.p12

eq1= p1.g.p1==(m^2)

Problem #3 :

a) CMS System

Clear["Global`*"]

g=DiagonalMatrix[{1,-1,-1,-1}];
g //MatrixForm

(* BEFORE *)
p1={e1,0,0,+pz};
p2={e2,0,0,-pz};
p12=p1+p2;
e1=e2=e; (* set energies equal *)

a) Fixed Target System

Clear["Global`*"]

g=DiagonalMatrix[{1,-1,-1,-1}];
g //MatrixForm

(* BEFORE *)
p1={e,0,0,+pz};
p2={m,0,0,0};
p12=p1+p2;

sBefore= p12.g.p12

eq1= p1.g.p1==(m^2)

Problem #5 :

a) z-Boost:

Clear["Global`*"]

boostz={{1,0,0,β},{0,1,0,0},{0,0,1,0},{β,0,0,1}};
boostz //MatrixForm

(* CMS *)
p1={e,0,0,+pz};
p2={e,0,0,-pz};

p2Boosted= boostz.p2

p2Boosted[[4]]==0

sol= Solve[p2Boosted[[4]]==0,β][[1]]

p1Boosted= boostz.p1

p1Boosted[[1]] /.sol

Problem #6 :

I' ve done this for a boost along the z direction .  You repeat for either an x or y boost .

Clear["Global`*"]

f={{0,-ex,-ey,-ez},{ex,0,-bz,+by},{ey,bz,0,-bx},{ez,-by,bx,0}};
f //MatrixForm

Transpose[f]+f

boostz={{1,0,0,β},{0,1,0,0},{0,0,1,0},{β,0,0,1}};
boostz //MatrixForm

boostz.f.Transpose[boostz] //MatrixForm