The paper represents an advancement of research the fundamental problem of which is a classification of algebras A (Weil algebras primarily) having a nontrivial fixed point subalgebra (with respect to \underlineall algebra automorphisms). The main result is the determination of the algebra order allowing a nontrivial fixed point subalgebra. Moreover, an autonomous importance of some results about socle elements of A and the unipotency of algebra automorphisms is highlighted.
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