X(a,b,c):=matrix(
[1,0,0],[0,1,0],[0,0,1],
[ sqrt(-b)/2,    sqrt(-a)/2,  0],
[ sqrt(-c)/2,    0,          sqrt(-a)/2],
[ 0,            sqrt(-c)/2,  sqrt(-b)/2],
[sqrt(b*c)/3,sqrt(a*c)/3,sqrt(a*b)/3])$
X(a,b,c):=matrix(
[1,0,0],[0,1,0],[0,0,1],
[ (%i*b)/2,    (%i*a)/2,  0],
[ (%i*c)/2,    0,          (%i*a)/2],
[ 0,            (%i*c)/2,  (%i*b)/2],
[(b*c)/3,(a*c)/3,(a*b)/3]);
v:matrix([a,b,c]);
u:matrix([x,y,z]);
vt:transpose(v)$
X(a,b,c).v;
MT:radcan(transpose(X(a,b,c)).X(a,b,c))$
metric_tensor=MT;
det=radcan(determinant(MT));
reduced=expand(
subst(a,a^2,subst(b,b^2,subst(c,c^2,
subst(a,a^3,subst(b,b^3,subst(c,c^3,
subst(a,a^4,subst(b,b^4,subst(c,c^4,
%)))))))));
norm=radcan(expand(vt.transpose(X(a,b,c)).X(a,b,c).v));
expand(subst(a,a^2,subst(b,b^2,subst(c,c^2,
subst(a,a^(3/2),subst(b,b^(3/2),subst(c,c^(3/2),
subst(a,a^(1/2),subst(b,b^(1/2),subst(c,c^(1/2),
subst(a*b,(a*b)^(1/2),subst(a*c,(a*c)^(1/2),subst(b*c,(b*c)^(1/2),
%)))))))))))))$
reduced=expand(subst(a,a^2,subst(b,b^2,subst(c,c^2,
subst(a,a^(3/2),subst(b,b^(3/2),subst(c,c^(3/2),
subst(a,a^(1/2),subst(b,b^(1/2),subst(c,c^(1/2),
subst(a*b,(a*b)^(1/2),subst(a*c,(a*c)^(1/2),subst(b*c,(b*c)^(1/2),
%)))))))))))));
inner_product=radcan(expand(vt.transpose(X(a,b,c)).X(x,y,z).u));
reduced=expand(subst(a,a^2,subst(b,b^2,subst(c,c^2,
subst(a,a^(3/2),subst(b,b^(3/2),subst(c,c^(3/2),
subst(a,a^(1/2),subst(b,b^(1/2),subst(c,c^(1/2),
subst(a*b,(a*b)^(1/2),subst(a*c,(a*c)^(1/2),subst(b*c,(b*c)^(1/2),
subst(x,x^2,subst(y,y^2,subst(z,z^2,
subst(x,x^(3/2),subst(y,y^(3/2),subst(z,z^(3/2),
subst(x,x^(1/2),subst(y,y^(1/2),subst(z,z^(1/2),
subst(x*y,(x*y)^(1/2),subst(x*z,(x*z)^(1/2),subst(y*z,(y*z)^(1/2),
(%) )))))))))))))))))))))))));