  In this article we construct and discuss several aspects of the two-component
spinorial formalism for six-dimensional spacetimes, in which chiral spinors are
represented by objects with two quaternionic components and the spin group is
identified with $SL(2;\mathbb{H})$, which is a double covering for the Lorentz
group in six dimensions. We present the fundamental representations of this
group and show how vectors, bivectors, and 3-vectors are represented in such
spinorial formalism. We also complexify the spacetime, so that other signatures
can be tackled. We argue that, in general, objects built from the tensor
products of the fundamental representations of $SL(2;\mathbb{H})$ do not carry
a representation of the group, due to the non-commutativity of the quaternions.
The Lie algebra of the spin group is obtained and its connection with the Lie
algebra of $SO(5,1)$ is presented, providing a physical interpretation for the
elements of $SL(2;\mathbb{H})$. Finally, we present a bridge between this
quaternionic spinorial formalism for six-dimensional spacetimes and the
four-component spinorial formalism over the complex field that comes from the
fact that the spin group in six-dimensional Euclidean spaces is given by
$SU(4)$.
