  Let $X$ be a Banach space of dimension $\geq 2$ over the real or complex
field ${\mathbb F}$ and
  ${\mathcal A}$ a standard operator algebra in ${\mathcal B}(X)$. A map
$\Phi:{\mathcal A} \rightarrow {\mathcal A}$ is said to be strong
$3$-commutativity preserving if $[\Phi(A),\Phi(B)]_3 = [A,B]_3$ for all $A,
B\in{\mathcal A}$, where $[A,B]_3$ is the 3-commutator of $A,B$ defined by
$[A,B]_3=[[[A,B],B],B]$. The main result in this paper is shown that, if $\Phi$
is a surjective map on ${\mathcal A}$, then $\Phi$ is strong $3$-commutativity
preserving if and only if there exist a functional $h :{\mathcal A} \rightarrow
{\mathbb F}$ and a scalar $\lambda \in{\mathbb F}$ with $\lambda^4 = 1$ such
that $\Phi(A) = \lambda A + h(A)I$ for all $A \in{\mathcal A}$.
