2ij

2ij is an algebraic expression that appears in quaternion mathematics, formed from the quaternion units i, j, and k. In the real quaternions H, the units satisfy i^2 = j^2 = k^2 = ijk = -1, and the multiplication rules ij = k, ji = -k, jk = i, kj = -i, ki = j, and ik = -j. Using these relations, ij equals k, so 2ij simplifies to 2k. Therefore, 2ij is a pure imaginary quaternion with zero scalar part and components (0, 0, 0, 2) in the basis {1, i, j, k}.

Quaternions are non-commutative, so the order matters: ji equals -k, and thus 2ji equals -2k, not 2k. The expression 2ij commonly appears in algebraic manipulations within quaternion calculus and can be relevant in contexts involving 3D rotations, where quaternions provide a framework for representing orientations.

As a quaternion, 2ij has magnitude 2 and lies along the k-axis in the imaginary subspace. While i, j, and k are often interpreted as orthogonal imaginary units, their use in quaternions encodes rotational information rather than ordinary 2D vector components.

In summary, 2ij denotes two times the product of i and j in quaternion algebra, which equals 2k under standard relations and reflects the non-commutative structure of quaternions.