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We prove that any Bernstein algebra $(A, \omega)$ is isomorphic to asemidirect product $V \ltimes_{(\cdot, \, \Omega)} \, k$ associated to acommutative algebra $(V, \cdot)$ such that $(x^2)^2 = 0$, for all $x\in A$ andan idempotent endomorphism $\Omega = \Omega^2 \in {\rm End}_k (V)$ of $V$satisfying two compatibility conditions. The set of types of $(1 +|I|)$-dimensional Bernstein algebras is parametrized by an explicitelyconstructed (using linear algebra tools) classified object. The automorphismsgroup of any Bernstein algebra is described as a subgroup of the canonicalsemidirect product of groups $(V, +) \ltimes {\rm GL}_k (V)$.

On the structure and classification of Bernstein algebras