TITLE: Why are There No "Triernions" (3-dimensional analogue of complex numbers / quaternions)?
QUESTION [118 upvotes]: Since there are  complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions"). 
Yet no one uses these. Why is this?

REPLY [12 votes]: The closest thing to triterniums would be the structure $[\mathbb R^3, +, \times]$ where $``\times"$ represents the cross product
$$(a_1 \mathbf i + b_1 \mathbf j + c_1 \mathbf k) \times 
  (a_2 \mathbf i + b_2 \mathbf j + c_2 \mathbf k) =
\left| \begin{matrix}
    \mathbf i & \mathbf j & \mathbf k \\
          a_1 &       b_1 &       c_1 \\
          a_2 &       b_2 &       c_2 \\
\end{matrix} \right|$$
It distributes over $``+"$,
is anti commutative,
and isn't associative, but
$[a \times (b \times c)] + [b \times (c \times a)] + [c \times (a \times b)] = 0$