Anticommutators:
{ 1*C^+(1,0) , 1*C(1,0) } = 1
{ 1*C^+(1,0) , 1*C(2,0) } = 0
{ 1*C^+(1,0) , 1*C(3,0) } = 0
{ 1*C^+(2,0) , 1*C(1,0) } = 0
{ 1*C^+(2,0) , 1*C(2,0) } = 1
{ 1*C^+(2,0) , 1*C(3,0) } = 0
{ 1*C^+(3,0) , 1*C(1,0) } = 0
{ 1*C^+(3,0) , 1*C(2,0) } = 0
{ 1*C^+(3,0) , 1*C(3,0) } = 1
Commutators:
[ 1*C^+(1,0) , 1*C(1,0) ] = -1 + 2*C^+(1,0)C(1,0)
[ 1*C^+(1,0) , 1*C(2,0) ] = 2*C^+(1,0)C(2,0)
[ 1*C^+(1,0) , 1*C(3,0) ] = 2*C^+(1,0)C(3,0)
[ 1*C^+(2,0) , 1*C(1,0) ] = 2*C^+(2,0)C(1,0)
[ 1*C^+(2,0) , 1*C(2,0) ] = -1 + 2*C^+(2,0)C(2,0)
[ 1*C^+(2,0) , 1*C(3,0) ] = 2*C^+(2,0)C(3,0)
[ 1*C^+(3,0) , 1*C(1,0) ] = 2*C^+(3,0)C(1,0)
[ 1*C^+(3,0) , 1*C(2,0) ] = 2*C^+(3,0)C(2,0)
[ 1*C^+(3,0) , 1*C(3,0) ] = -1 + 2*C^+(3,0)C(3,0)

Algebra:
x = 1*C(0,0)
y = 1*C^+(1,0)
-x = -1*C(0,0)
x + 2.0 = 2 + 1*C(0,0)
2.0 + x = 2 + 1*C(0,0)
x - 2.0 = -2 + 1*C(0,0)
2.0 - x = 2 + -1*C(0,0)
3.0*y = 3*C^+(1,0)
y*3.0 = 3*C^+(1,0)
x + y = 1*C^+(1,0) + 1*C(0,0)
x - y = -1*C^+(1,0) + 1*C(0,0)
(x + y)*(x - y) = 2*C^+(1,0)C(0,0)

N^3:
N = 1*C^+(0,dn)C(0,dn) + 1*C^+(0,up)C(0,up)
N^3 = 1*C^+(0,dn)C(0,dn) + 1*C^+(0,up)C(0,up) + 6*C^+(0,dn)C^+(0,up)C(0,up)C(0,dn)