  Let $K$ be an algebraically closed field of characteristic $0$, and let
$H^{\mu}$ denote the Hilbert scheme of $\mu$ points of the affine space $A^n$.
An elementary component $E$ of $H^{\mu}$ is an irreducible component such that
every $K$-point $[I]$ $\in$ $E$ represents a length-$\mu$ closed subscheme
Spec$(K[x_1,\dots,x_n]/I)$ $\subseteq$ $A^n$ that is supported at one point. In
a previous article we found some new examples of elementary components; in this
article, we simplify the methods and extend the range of the previous paper to
find several more examples. In addition, we present a "plausibility test" that
suggests the existence of a vast number of similar examples.
