Definition of ImaginaryPart for quaternion algebras unclear

[fingolfin]

From the manual:

For a quaternion z = c_1 e + c_2 i + c_3 j + c_4 k, ComplexConjugate returns c_1 e - c_2 i - c_3 j - c_4 k, [...]
When ComplexConjugate is defined for an object z then RealPart and ImaginaryPart return (z + ComplexConjugate( z )) / 2 and (z - ComplexConjugate( z )) / 2 i, respectively, where i denotes the corresponding imaginary unit.

But what is the "corresponding imaginary unit" ?

The code in lib/algsc.gi does not clarify, to the contrary, I don't understand how it is supposed to be correct. It says this:

InstallMethod( ImaginaryPart,
    "for a quaternion",
    [ IsQuaternion and IsSCAlgebraObj ],
    function( quat )
    local v, z;

    v:= ExtRepOfObj( quat );
    z:= Zero( v[1] );
    return ObjByExtRep( FamilyObj( quat ), [ v[2], z, v[4], -v[3] ] );
    end );

That looks to me as if it can only be valid for a single specific set of parameters a,b, not in general. But note this:

gap> A:= QuaternionAlgebra( Rationals );
<algebra-with-one of dimension 4 over Rationals>
gap> AssignGeneratorVariables(A);
#I  Assigned the global variables [ e, i, j, k ]
gap> v:=e+i+j+k;
e+i+j+k
gap> (v - ComplexConjugate( v )) / (2*i);
e+(-1)*j+k
gap> ImaginaryPart(v);
e+j+(-1)*k

Perhaps @ThomasBreuer has some insights on this?

[fingolfin]

@ThomasBreuer I think you wrote some of that code, do you have any insights on it?

[ThomasBreuer]

• Yes, there is an error.
• I will prepare a pull request proposing a fix.
• (I think that RealPart and ImaginaryPart are not really useful functions.)