  In this paper, a simple Lie algebra, referred to as the completed Witt Lie
algebra, is introduced. Its derivation algebra and automorphism group are
completely described. As a byproduct, it is obtained that the first cohomology
group of this Lie algebra with coefficients in its adjoint module is trivial.
Furthermore, we completely determine the conjugate classes of this Lie algebra
under its automorphism group, and also obtain that this Lie algebra does not
contain any nonzero ad-locally finite element.
