Definition. Hamilton's Quaternions [fgap-0006]
Definition. Hamilton's Quaternions [fgap-0006]
The algebra of real quaternions is the 4-dimensional real vector space \[ \mathbb {H} =\mathbb {R}1\oplus \mathbb {R}i\oplus \mathbb {R}j\oplus \mathbb {R}k. \] Thus every quaternion can be written uniquely as \[ q=a+bi+cj+dk,\qquad a,b,c,d\in \mathbb {R}. \] Addition and real scalar multiplication are coordinatewise. Multiplication is extended bilinearly from \(1\) as the multiplicative identity and the table \[ \begin {gathered} i^2=j^2=k^2=ijk=-1,\\ ij=k,\quad jk=i,\quad ki=j,\\ ji=-k,\quad kj=-i,\quad ik=-j. \end {gathered} \] This makes \(\mathbb {H}\) an associative, noncommutative real algebra. In Voight's notation, \(\mathbb {H}=(-1,-1\mid \mathbb {R})\); see [voight2021quaternion, def. 2.2.1 and ex. 2.2.3].
More generally, when \(F\) is a field of characteristic different from \(2\) and \(a,b\in F^\times \), the symbol \[ (a,b\mid F) \] denotes the quaternion algebra over \(F\) generated by elements \(i,j\) with \[ i^2=a,\qquad j^2=b,\qquad ji=-ij. \] The two slots record the scalars assigned to the squares of the two generators. With \(k=ij\), the elements \(1,i,j,k\) form an \(F\)-basis. Thus the two entries \(-1,-1\) in \((-1,-1\mid \mathbb {R})\) say precisely that \(i^2=-1\) and \(j^2=-1\); the anticommutation relation then gives \(k^2=-1\).
For \(q=a+bi+cj+dk\), its quaternion conjugate and reduced norm are \[ \overline {q}=a-bi-cj-dk,\qquad \operatorname {nrd}(q)=q\overline {q} =a^2+b^2+c^2+d^2. \] We also have \(\overline {q}q=\operatorname {nrd}(q)\). If \(q\ne 0\), then \(\operatorname {nrd}(q)>0\) and \[ q^{-1}=\frac {\overline {q}}{\operatorname {nrd}(q)}. \] So every nonzero quaternion is invertible. This is the standard involution and reduced norm specialized to Hamilton's Quaternions; see [voight2021quaternion, secs. 3.1--3.3].
The corresponding Mathlib type is Quaternion.