box% sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 9.1, Release Date: 2020-05-20                     │
│ Using Python 3.8.5. Type "help()" for help.                        │
└────────────────────────────────────────────────────────────────────┘
sage: R.<a,b,c,d>=PolynomialRing(QQ)
sage: ideal(a^2+b^2-1,c^2+d^2-1,(a*c-b*d)-1,(a*d+b*c)-0).groebner_basis()
[c^2 + d^2 - 1, a - c, b + d]
sage: R.<a,b,c,d,e,f>=PolynomialRing(QQ)
sage: for g in ideal(a^2+b^2-1,c^2+d^2-1,e^2+f^2-1,((a*c-b*d)*e-(a*d+b*c)*f)-1,((a*d+b*c)*f+(a*d+b*c)*e)-0).groebner_basis(): print(g)
a*b^5 - a*b^3*d^2 + 2*b^4*d^2 - a*b*d^4 - 2*b^2*d^4 - c*d^5 - 1/2*a*b^3 - b^4 + 3/2*a*b*d^2 + c*d^3 + d^4 - a*d*e - 1/2*b*d*e - 3/4*b*c*f - 3/4*a*d*f - 5/8*a*b + 3/8*b^2 + 3/8*c*d - 7/8*d^2
b^6 - 2*a*b^3*d^2 - b^4*d^2 + 2*a*b*d^4 - b^2*d^4 + d^6 + a*b^3 - b^4 - a*b*d^2 + 2*b^2*d^2 - d^4 - 1/4*b^2 + 1/4*d^2
a*b*d^3*f + c*d^4*f - 1/4*a*b^4 + 1/4*b^5 - 1/2*a*b^2*d^2 - b^3*d^2 + 3/4*a*d^4 + 3/4*b*d^4 - 3/2*a*b*d*f - 3/2*c*d^2*f + 5/8*a*b^2 + 3/8*b^3 - 5/8*a*d^2 - 3/8*b*d^2 + 1/2*b*f^2 - 1/2*c*e - 3/4*d*e + 1/2*a - 3/4*b
b^2*d^3*f - d^5*f - 1/4*a*b^4 - 1/4*b^5 + a*b^2*d^2 - 1/2*b^3*d^2 - 3/4*a*d^4 + 3/4*b*d^4 - 1/8*a*b^2 + 3/8*b^3 + 1/8*a*d^2 - 3/8*b*d^2
a*b^3*f - a*b*d^2*f - b^2*d^2*f + d^4*f - a*b^2*d + b^3*d + a*d^3 - b*d^3 - 1/2*b^2*f + 1/2*d^2*f
b^4*f + a*b*d^2*f - b^2*d^2*f + c*d^3*f - a*b^2*d - b^3*d + a*d^3 + b*d^3 + 2*a*d*f^2 - 1/2*a*b*f - 3/2*b^2*f - 1/2*c*d*f + 3/2*d^2*f - 1/4*a*c - 3/4*b*c - 3/4*a*d + 1/4*b*d + 1/4*e
a*b^2*d*f - a*d^3*f - 1/4*a*b^3 - 1/4*b^4 + 1/2*b^2*d^2 - 1/4*c*d^3 - 1/4*d^4 - 1/2*a*d*e - 1/4*b*c*f - 1/4*a*d*f - 1/8*a*b + 1/4*b^2 + 3/8*c*d - 1/4*d^2 - 1/8*f^2
b^3*d*f - b*d^3*f + 1/4*a*b^3 - 1/4*b^4 - 1/2*a*b*d^2 - 1/4*c*d^3 + 1/4*d^4 - 1/2*a*d*e - 1/4*b*c*f - 1/4*a*d*f - 1/8*a*b + 3/8*c*d - 1/8*f^2
a*b*f^2 - a*d*e - 1/2*b*c*f - 1/2*a*d*f - 3/4*a*b - 1/4*b^2 + 1/4*c*d + 1/4*d^2
b^2*f^2 - b*d*e - 1/2*b*c*f - 1/2*a*d*f + 1/4*a*b - 3/4*b^2 + 1/4*c*d - 1/4*d^2 - 1/2*f^2
b*c*f^2 + a*d*f^2 - 1/2*b*c - 1/2*a*d
c*d*f^2 + a*d*e + 1/2*b*c*f + 1/2*a*d*f + 1/4*a*b + 1/4*b^2 - 3/4*c*d - 1/4*d^2
d^2*f^2 - b*d*e - 1/2*b*c*f - 1/2*a*d*f + 1/4*a*b - 1/4*b^2 + 1/4*c*d - 3/4*d^2 - 1/2*f^2
a*b*c - b^2*d - c*f^2 + d*f^2 - a*e - b*e + c
b^2*c + a*b*d + c*f^2 + d*f^2 + a*e - b*e - c - d
a*c*d - b*d^2 - a*f^2 + b*f^2 - c*e - d*e + a
b*c*d + a*d^2 + a*f^2 + b*f^2 + c*e - d*e - a - b
a*b*e + 1/2*a*b*f - 1/2*b^2*f + 1/2*c*d*f + 1/2*d^2*f - 1/2*b*c + 1/2*a*d
b^2*e + 1/2*a*b*f + 1/2*b^2*f + 1/2*c*d*f - 1/2*d^2*f + 1/2*a*c + 1/2*b*d - 1/2*e
a*c*e - b*d*e - b*c*f - a*d*f - 1
b*c*e + a*d*e + b*c*f + a*d*f
c*d*e + 1/2*a*b*f + 1/2*b^2*f + 1/2*c*d*f - 1/2*d^2*f + 1/2*b*c - 1/2*a*d
d^2*e + 1/2*a*b*f - 1/2*b^2*f + 1/2*c*d*f + 1/2*d^2*f + 1/2*a*c + 1/2*b*d - 1/2*e
a*c*f + b*c*f + a*d*f - b*d*f - e*f + f^2
a*e*f - a*f^2 - c*e - c*f + a
b*e*f - b*f^2 + d*e + d*f + b
c*e*f - c*f^2 - a*e - a*f + c
d*e*f - d*f^2 + b*e + b*f + d
e*f^2 - 1/2*a*c - 1/2*b*c - 1/2*a*d + 1/2*b*d + 1/2*e
f^3 + 1/2*a*c + 1/2*b*c + 1/2*a*d - 1/2*b*d - 1/2*e
a^2 + b^2 - 1
c^2 + d^2 - 1
e^2 + f^2 - 1
sage: