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I am reading about quaternions and their basic properties from this article Algebra and Geometry of Hamilton’s Quaternions

quaternions are quadruples of real numbers $q = (α, β, γ, δ)$ forming a four dimensional real vector space $H$ (named after Hamilton). In terms of the canonical basis: $e = (1, 0, 0, 0), i= (0, 1, 0, 0), j= (0, 0, 1, 0)$ and $k = (0, 0, 0, 1) (3)$ ....

Since any quaternion is a linear combination of the basis quaternions $e, i, j$ and $k$, it suffices to specify their products. First, $e$ is taken to be the multiplicative identity (sometimes denoted $1$), so $eq = qe = q$ for any quaternion $q$. In addition, $i^2 = j^2 = k^2 = −e$, while $ij = −ji = k$, $jk = −kj = i$ and $ki = −ik = j. (4)$

However I do not understand how to obtain the eqns $(4)$ using the forms of $e,i,j,k$ given in $(3)$? Any clue regarding the calculations would be helpful.

Anindita Sarkar
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    From the possible representation (3) you cannot obtain (4). To do that remember that the vector space $H$ can also be seen as being spanned by three rotation matrices plus the identity matrix. – Kurt G. Sep 11 '23 at 17:40
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    "$e$ is taken to be ..." suggests $(4)$ is a list of definitions, and formula $(5)$ follows from them. – user170231 Sep 11 '23 at 17:41
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    (4) is part of the definition. It doesn't follow from (3). – Karl Sep 11 '23 at 17:52
  • $(4)$ is part of definition. But if $i,j,k$ as given in $(3)$ are part of quaternion then they all should satisfy $(4)$, right? – Anindita Sarkar Sep 11 '23 at 18:19
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    (3) doesn't say anything about multiplication whatsoever, it just names the standard basis of $\mathbb{R}^4$. (4) is a statement with new content, it is telling you how to define a new multiplication operation. – Qiaochu Yuan Sep 11 '23 at 18:59

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