Let F be a field and L a loop such that FL is an alternative algebra.
We denote by U(FL) the Moufang loop of invertible elements in FL. We s
ay that U(FL) is FC if, for every unit u is an element of U(FL), the s
et {x(-1)ux \ x is an element of U(FL)} is finite. In this paper, we f
ind necessary and sufficient conditions for U(FL) to be FC.