result

By substituting psieta and hinyphi for psi[theta] and h[y] respectively in foo51=log f(y;θ), we obtain the canonical form of log f(y;η) as follows.

This is the canonical form  of log f(y;η).

-(dim*Log[2*Pi])/2 - (ry[l1]*ry[u1])/2 + ry[l1]*se[u1] - 
 (se[l1]*se[u1])/2 + (o*ry[l1]*tp3[l2, u1, u2])/2 - 
 (o*ry[l1]*ry[l2]*ry[l3]*tp3[u1, u2, u3])/6 + 
 (o*ry[l1]*se[l2]*se[l3]*tp3[u1, u2, u3])/2 - 
 (o*se[l1]*se[l2]*se[l3]*tp3[u1, u2, u3])/3 + 
 (o^2*tp3[l1, l2, l3]*tp3[u1, u2, u3])/6 - 
 (o^2*ry[l1]*ry[l2]*tp3[l3, l4, u1]*tp3[u2, u3, u4])/4 - 
 (o^2*tp4[l1, l2, u1, u2])/8 + 
 (o^2*ry[l1]*ry[l2]*tp4[l3, u1, u2, u3])/4 - 
 (o^2*ry[l1]*ry[l2]*ry[l3]*ry[l4]*tp4[u1, u2, u3, u4])/24 + 
 (o^2*ry[l1]*se[l2]*se[l3]*se[l4]*tp4[u1, u2, u3, u4])/6 - 
 (o^2*se[l1]*se[l2]*se[l3]*se[l4]*tp4[u1, u2, u3, u4])/8

We also show the metric by using rulephi2, and name it "phi2eta" for later use.

Kdelta[ua, ub] + o*se[l1]*tp3[u1, ua, ub] + 
 (o^2*se[l1]*se[l2]*tp4[u1, u2, ua, ub])/2