Quaternion Basics

Algebra Level 3

Let \(A\), \(B\) and \(C\) be quaternions whose values are

\[ A = 2 - i + j - k \]

\[ B = 3 + i - j + k \]

\[ C = 1 + i + j + k \]

with \(i\), \(j\), and \(k\) as the quaternion units following the property

\[ i^2 = j^2 = k^2 = ijk = -1 \]

If \( Q = \large \sum\limits_{cyc} ABC \) and is expressed in the form \( w + xi + yj + zk\), with \(w\), \(x\), \(y\), and \(z\) being integers, find \(w+x+y+z\).

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