1 | initial version |

The reason is

```
TypeError: base ring of quaternion algebra must be a field
```

and the documentation of `QuaternionAlgebra`

mentions it must be a field.
To avoid the reason one may go to the source directly, this gives the short solution:

**Short answer:**

```
R.<e> = PolynomialRing(QQ, names='E').quotient( E^2 )
D, (i,j,k) = sage.algebras.free_algebra_quotient.hamilton_quatalg( R )
```

Test:

```
sage: (1+2*i+3*j+4*k+5*e)^6
(117480*e-12208) + (33360*e+8944)*i + (50040*e+13416)*j + (66720*e+17888)*k
sage: F.<I,J,K> = QuaternionAlgebra(QQ, -1,-1)
sage: (1+2*I+3*J+4*K)^6
-12208 + 8944*I + 13416*J + 17888*K
```

**Explicit answer**
A more elaboarted way would be:

```
S.<E> = PolynomialRing( QQ )
R.<e> = S.quotient( E^2 )
A = FreeAlgebra(R, 3, 'u')
U = A.monoid()
I, J, K = U.gens()
mons = [ U(1), I, J, K, ]
M = MatrixSpace(QQ, len(mons))
# columns for mats
# 1 I J K
mats = [ M( [ 0, 1, 0, 0, # 1*I
-1, 0, 0, 0, # I*I
0, 0, 0,-1, # J*I
0, 0, 1, 0, # K*I
] ),
M( [ 0, 0, 1, 0, # 1*J
0, 0, 0, 1, # I*J
-1, 0, 0, 0, # J*J
0,-1, 0, 0, # K*J
] ),
M( [ 0, 0, 0, 1, # 1*K
0, 0,-1, 0, # I*K
0, 1, 0, 0, # J*K
-1, 0, 0, 0, # K*K
] ),
]
D.<i,j,k> = FreeAlgebraQuotient(A, mons, mats)
```

(It would be maybe a good idea to mention in the documentation that the matrices correspond to the right multiplication with the generators, a while i was getting confused about this point.) Same test:

```
sage: (1+2*i+3*j+4*k+5*e)^6
(117480*e-12208) + (33360*e+8944)*i + (50040*e+13416)*j + (66720*e+17888)*k
```

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