  A purely local approach has been developed by Krishnamurthy and Kutzko to
compute Langlands-Shahidi local coefficient for ${\rm SL}(2)$ via types and
covers \`{a} la Bushnell-Kutzko. In this paper, we extend their method to the
non-split case and complete their project. We also study the algebraic
structure of Gelfand-Graev representations, which generalizes the results of
Chan-Savin and Mishra-Pattanayak to ${\rm SL}(2)$ over non-archimidean local
fields without any restriction on the characteristic.
