Let W [ x ± 1 , the E ^{ ± x < / sup > ] is, in the case of an indeterminate generalized Witt algebra constructed with the exponential function , and W [ X , and e x < / sup >] , W [x, e ± x ] and W [ x ± 1 , e x < / sup >] its subalgebra Kawamoto et al . determine W [ x, e < automorphism group sup> x ] W [ x, e ± x < / sup >] in this article , we determine the other two Lie algebra W [ x ± 1 e x ] and W [ x ± 1 , the automorphism group of e ± x < / sup >] , and the four Lee algebras and W [x, e x ] W [ x, e ± x < / sup >] - the main results are as follows : Let W be any four Lie algebra cohomology group a , denoted by Aut ( W) W automorphism group Aut (W [x ± 1, e x ]) ~ = F? , and Aut ( W [ x ± 1 , and e ± x < / sup>]) ~ = Z/2Z F . denoted Der (W) and Inn (W) W derivations and inner derivation space Der (W) = Inn (W
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