real=Element[{f,g,h},Reals]
B={{f,g},{0,h}};
B//MatrixForm
T=B\[Transpose].B;
T//MatrixForm
PKT[y_]=Collect[CharacteristicPolynomial[T,y],y];
PKT[x]

rl=l-> (Abs[f]-Abs[h])/Abs[f];
rm=m->g/f;
irm=g->m f;
rt=t->2-l;
rs=s->Sqrt[t^2+m^2];
rr =r->Sqrt[l^2+m^2];
ra=a->(s+r)/2;
rsig={s0->Abs[h]/a,s1->Abs[f]a};
irsig={h^2->(a s0)^2,f^2->(s1/a)^2};
rall=Flatten[{rsig,ra,rr,rs,rt,rm,rl}];
"OK"

Simplify[PKT[s0^2//.rall]==0,Assumptions->real]
Simplify[PKT[s1^2//.rall]==0,Assumptions->real]

τnum=(m/(s+t)+m/(r+l))(1+a);
τ=-2/m(1-a^2);
FullSimplify[(τ-τnum)//.{ra,rr,rs,rt},Assumptions->Element[{l,m},Reals]]

ClearAll[cv,sv]
rv1={cv->2/Sqrt[τ^2+4],sv->τ/Sqrt[τ^2+4]};
v1={cv,sv};
Tm=Simplify[(T-s1^2 IdentityMatrix[2])//.Flatten[{irm,irsig}]];
Tm//MatrixForm;
FullSimplify[((Tm.v1)[[1]]/.rv1)==0,Assumptions->{a>0,m ϵ Reals,s0>0,s1>0}]

rsign={x_/Abs[x_]:>ϵ[x]};

u1={cu,su}
Simplify[Solve[B.v1==s1 u1,{cu,su}]/.Flatten[{rsig,irm}]]/.rsign