  The purpose of this note is to describe when a general complex algebraic
$^*$-algebra is pre-$C^*$-normed, and to investigate their structure when the
$^*$-algebras are Baer $^*$-rings in addition to algebraicity. As a main result
we prove the following theorem for complex algebraic Baer $^*$-algebras: every
$^*$-algebra of this kind can be decomposed as a direct sum $M\oplus B$, where
$M$ is a finite dimensional Baer $^*$-algebra and $B$ is a commutative
algebraic Baer $^*$-algebra. The summand $M$ is $^*$-isomorphic to a finite
direct sum of full complex matrix algebras of size at least $2\times2$. The
commutative summand $B$ is $^*$-isomorphic to the linear span of the
characteristic functions of the clopen sets in a Stonean topological space.
  As an application we show that a group $G$ is finite exactly when the complex
group algebra $\mathbb{C}[G]$ is an algebraic Baer $^*$-algebra.
