Pure quaternion - Knowlpedia

A pure quaternion in a quaternion algebra $B$ over a field $F$ of characteristic different from two is an element with reduced trace zero. Their space is

B_0=\ker(\operatorname{trd}:B\to F) =\{x\in B:\bar x=-x\}.

It has dimension three. In $B=(a,b)_F$ , it is $Fi\oplus Fj\oplus Fij$ , and $B=F1\oplus B_0$ .

Squares and the norm

For $x\in B_0$ , the reduced characteristic identity gives $x^2=-\operatorname{nrd}(x)$ . In the rational Hamilton algebra,

(bi+cj+dk)^2=-(b^2+c^2+d^2).

This is why quadratic-field embeddings there connect to sums of three squares.

Linear space and Lie algebra

Pure quaternions are not generally closed under multiplication: $i^2=a$ is scalar. They are closed under the commutator $[x,y]=xy-yx$ , because reduced trace is cyclic. The multiplication and the commutator are different algebraic structures.

References

John Voight, Quaternion Algebras, author's text, §§2.4 and 5.1, the trace-zero subspace.
The coordinate formulas and reduced characteristic identity directly verify the displayed properties.