Lie rings admitting an automorphism of order 4 with few fixed points. II

We consider a Lie ring (algebra) L that admits an automorphism φ of order 4 with a finite number m of fixed points (with a fixed-point subalgebra of finite dimension m). It is proved that L contains a subring S of m-bounded index in the additive group L (a subalgebra S of m-bounded codimension), which possesses a nilpotent ideal I of class bounded by some constant, such that the factor-ring S/I is nilpotent of class ≤2. As a consequence, it is proved that, under the same conditions, L has a subring G of m-bounded index in the additive group of L (a subalgebra G of m-bounded codimension), in which an ideal generated by the Lie subring G, ?²]=«ng?g+g^{? 2} | g∈G»ng (the subalgebra G, ?²]=«ng?g+g^{? 2} | g∈G»ng is an ideal in G which) is nilpotent of class bounded by some constant (and its factor-algebra G/G, ?²] is nilpotent of class ≤2 with a derived algebra (square) of m-bounded dimension). In proofs, we use the results of 1] and develop further the version of the method of generalized centralizers employed therein.