TITLE: Is every $A \in \mathrm{SL}_n(\mathbb C)$ a product of four unipotent matrices?
QUESTION [11 upvotes]: Is every matrix $A \in \mathrm{SL}_n(\mathbb C)$ a product of four unipotent matrices?
I have verified that this is true if $n = 2$, and I believe I have came across this result before. However, I cannot find a reference to this.

REPLY [11 votes]: If you only need four factors then you can have much more than that. Namely, you can take as the unipotent matrices upper and lower uni-triangular matrices. It was noted by G. Strang in "Every unit matrix is a LULU" that for any field $K$ anfy $A\in M(n,K)$ can be presented as $A=U_1L_1U_2L_2$, where $U_i$ are upper uni-triangular and $L_i$ are lower uni-triangular.
In fact, this is true not only for fields, but also for rings of stable rank $1$ (in particular, for semilocal rings, for boolean rings, for the ring of all entire functions and for the ring of algebraic integers), see this paper (the result was known to the experts long before that, but mostly in terms of triangular decompositions).
As noted in the answer by Gjergji Zaimi, there is a paper by A. Sourour, where he proved that over a field $F$ there is a presentation as a product of three unipotent matrices (two for non-central elements). This can not be carried to the decompositions in terms of lower and upper uni-tringular marices. Namely, every matrix $A\in \operatorname{SL}(n, R)$ is a product $U_1LU_2$ if and only if $R$ is a boolean ring (reference). However, T. Toffoli proved that almost all (w.r.t. Lebesgue measure) elements of $\operatorname{SL}(n,\mathbb{R})$ are ULU, and that for any $A$ there exists a permutation matrix $P$ such that either $PA$ or $-PA$ is ULU.