Let $ \mathbb{H}$ be a division algebra of quaternions.

**Is there an element in $M ( \mathbb{H} ) $ that cannot be written as the left
multiplication map and right multiplication map**? 

Let $D$ be an algebra. For $a, b \in D $, maps $L\_{a}, R\_{b} : D \longrightarrow D$ are the left multiplication map and right multiplication map, by

$L\_{a} (x) := ax$

$ R\_{b} (x) := xb$.