  In these notes we study left-invariant involutive structures on
$\mathrm{SU}(2)$, the most na\"ive non-commutative compact Lie group. We
determine closedness of the range (in the smooth topology) of a single complex
vector field spanning the standard CR structure of $\mathrm{SU}(2)$ and also
compute the smooth cohomology spaces of a corank $1$ structure. In our
approach, it is fundamental to understand concretely the irreducible
representations of the ambient Lie group and how left-invariant vector fields
operate on their matrix coefficients (which we borrow from the book of
Ruzhansky and Turunen (2010)).
  Our purpose is solely to provide some easy applications of the theory
developed in a previous paper (2019), as the results shown here can probably be
obtained by more direct methods.
