  Let $A$ be a unital dense algebra of linear mappings on a complex vector
space $X$. Let $\phi=\sum_{i=1}^n M_{a_i,b_i}$ be a locally quasi-nilpotent
elementary operator of length $n$ on $A$. We show that, if $\{a_1,\ldots,a_n\}$
is locally linearly independent, then the local dimension of
$V(\phi)=\spa\{b_ia_j: 1 \leq i,j \leq n\}$ is at most $\frac{n(n-1)}{2}$. If
$\lDim V(\phi)=\frac{n(n-1)}{2} $, then there exists a representation of $\phi$
as $\phi=\sum_{i=1}^n M_{u_i,v_i}$ with $v_iu_j=0$ for $i\geq j$. Moreover, we
give a complete characterization of locally quasi-nilpotent elementary
operators of length 3.
