  We study the complexity of horizontality in the twistor space $\hat{E}$
associated with an oriented vector bundle $E$ of rank $4$ with a
positive-definite metric over a torus. If the horizontality has finite
complexity of degree $d>2$ for an element of a fiber of $\hat{E}$, then the
complexity is expressed in terms of a finite subgroup of $SO(3)$ ([2]). In the
present paper, we observe that if the horizontality has infinite complexity
derived from one of the cases studied in [2], then the complexity is expressed
by a dense subset of $S^2$.
