TITLE: Is the Amitsur-Levitzki identity essentially unique?
QUESTION [15 upvotes]: Let us consider the matrix algebra. $Mat_n(\mathbb{C})$. The Amitsur-Levitzki identity states that for any matrices $X_1, X_2, ..., X_{2n} \in Mat_n(\mathbb{C})$ the sum $\Sigma_{\sigma \in S_{2n}} sgn(\sigma)X_{\sigma(1)}...X_{\sigma(2n)}$ vanishes identically.
Is it true that any identity (noncommutative polynomial, which always vanishes) between the matrices from $Mat_n(\mathbb{C})$ follows from Amitsur-Levitzki in some sense$^*$?
$*$ - I assume the following precise meaning: let us consider the oper of "noncommutative polynomial mappings" with the substitution as the composition of operations. It naturally (also non-linearly) acts on any associative algebra (as we can evaluate an $r$-tuple noncommutative polynomial of $k$ variables on the $k$-tuple of elements of an algebra to obtain an $r$-tuple). Is it true that the polynomials that act trivially on the $Mat_n(\mathbb{C})$ are precisely the ideal, generated by the Amitsur-Levitski polynomial?

REPLY [3 votes]: As mentioned before (for $2\times 2$ matrices), the answer is "no" .
Here I give some reference on the general case.
The identity of algebraicity is defined as:
$$ a_n(x,y_1,\dots, y_n)= \displaystyle \sum _{\sigma \in S_{n+1}} (-1)^\sigma x^{\sigma(0)}y_1 x^{\sigma(1)}y_2 \cdots x^{\sigma(n-1)}y_n x^{\sigma(n)} $$
In boook [1] we have:
Exercise 7.1.12
Show that the identity of algebraicity for $M_k(K)$, $k > 1$, does not follow from the standard identity $St_{2k}$.
In the book there is a hint for the exercise, but the original references for that are preprint [2] and paper [3].
Also in paper [4] there are other polynomial identities for $M_3(K)$ which does not follow from $St_6$ and $a_3$. 
There you also find a good list of references on the subject.
References:
[1] V. Drensky, Free Algebras and PI-algebras: Graduate Course in Algebra, Springer, Singapore, 1999.
[2] G. M. Bergrnan, Wild automorphisms offree P.I. algebras and some new identities, Preprint, Berkeley (1981). https://math.berkeley.edu/~gbergman/papers/unpub/wild_aut.pdf
[3] V. S. Drensky, A.K. Kasparian, Some polynomial identities of matrix algebras, C. R. Acad. Bulg. Sci. 36 (1983), 565-568.
[4] M. Domokos (1995) New identities for 3 × 3 Matrices, Linear and Multilinear Algebra, 38:3, 207-213, http://dx.doi.org/10.1080/03081089508818356