Let $S=\{AB-BA| A,B \in M_n(K)\}$ where $K$ is a field. Prove that $S$ is closed under matrix addition. I know there is a result that $S=$ collection of all trace $0$ matrices, and that collection forms a vector space. But I want to prove it independently i.e. 
For any $A,B,C,D\in M_n(K)$ we have to find $E,F\in M_n(K)$ such that $(AB-BA)+(CD-DC)=EF-FE$.
But I don't know how to solve this. But the thing I can observe that the above equation gives rise to $n^2$ equation (equating each entries of matrices of both the sides) with $2n^2$ variables (Total number of entries of $E,F$ is $2n^2$).Can this problem be simplified if we choose special kind of $E$ say diagonal matrix. 
Edit-(This is valid only for $\Bbb{R}$ or $\Bbb{C}$) $AB-BA=(A+aI)(B+bI)-(B+bI)(A+aI)$ for all $a,b\in\Bbb{R}$. And there is $a,b$ in $\Bbb{R}$ such that $A+aI,B+bI$ are invertible.
Hence, we can assume $A,B$ to be invertible matrices i.e. $S=\{AB-BA|A,B\in GL_n(K)\}$
 A: I have commented that fixing $E$ to have diagonal form would be too restrictive
because then $EF - FE$ would have only zeros on its main diagonal.
However, @ancientmathematician's comment
let me track down the original proof by Shoda (1936). As it turns out,
Shoda did use the idea with a diagonal $E$ after observing that similarity
transformations carry over to commutators, and in characteristic 0
a zero-trace matrix is similar to a matrix with zero main diagonal.
For the latter, Shoda referred to matrix normal forms. Actually
we do not need that much normalization, so I'll try an elementary proof here.
Lemma: Let $K$ be a field of characteristic 0,
$n$ a positive integer and $M\in\operatorname{Mat}_n(K)$
with $\operatorname{tr} M = 0$.
Then there exists $T\in\operatorname{SL}_n(K)$ such that
$T M T^{-1}$ has only zeros on its main diagonal.
Proof: For $n = 1$, $\operatorname{tr} M = 0$ implies $M = ((0))$,
and $T = I$ (identity matrix) works.
Now assume $n > 1$. We repeat the following sequence of steps,
each iteration finding a transformation $S\in\operatorname{SL}_n(K)$ such that
replacing $M$ by $S M S^{-1}$ increases the number of zero main
diagonal elements of $M$.
Multiplying all used $S$ (later ones at the left)
results in a suitable transformation $T$.

*

*If $M$ has only zeros on its main diagonal, $S = I$ works, and we are done.


*Otherwise, $M$ must have at least two nonzero main diagonal elements
$m_{ii}, m_{jj}$ with $1\leq i < j\leq n$
because if there were only one, it would equal the trace,
which is zero, contradiction.
Furthermore, not all nonzero main diagonal elements can be equal because
otherwise the zero-trace requirement would imply that their number is
a multiple of the field characteristic, but that's zero, contradiction.
Therefore we can choose $i,j$ such that $0\neq m_{ii}\neq m_{jj}\neq 0$.
We now define a transformation $S\in\operatorname{SL}_n(K)$ that differs from
the identity matrix only in its principal submatrix with rows/columns $i,j$:
$$\begin{pmatrix}s_{ii} & s_{ij}\\s_{ji} & s_{jj}\end{pmatrix}
= \begin{cases}
    \begin{pmatrix}1 & 0\\\frac{m_{ii}}{m_{ij}} & 1\end{pmatrix}
    & \text{if $m_{ij}\neq 0$, else}
\\  \begin{pmatrix}1 & -\frac{m_{ii}}{m_{ji}}\\0 & 1\end{pmatrix}
    & \text{if $m_{ji}\neq 0$, else}
\\  \begin{pmatrix}1 & \frac{m_{ii}}{m_{jj}-m_{ii}}
    \\ 1 & \frac{m_{jj}}{m_{jj}-m_{ii}}\end{pmatrix}
    & \text{if $m_{ij} = 0 = m_{ji}$}
\end{cases}$$
Then $M' = S M S^{-1}$ has the same main diagonal elements as $M$
except that $m_{ii}' = 0$ and $m_{jj}' = m_{ii} + m_{jj}$, preserving the
trace.
Off-diagonal elements with row or column index in $\{i,j\}$ may have changed
as well, but that does not matter.
Armed with the above Lemma, we can now assume $M$ to have an all-zero main
diagonal.
Fix $E\in\operatorname{Mat}_n(K)$ to be a diagonal matrix with pairwise distinct diagonal elements. Characteristic 0 ensures their existence.
To obtain $F$, let
$$f_{ij} = \begin{cases}
\frac{m_{ij}}{e_{ii}-e_{jj}}    & \text{for $i\neq j$}
\\  \text{(arbitrary)}              & \text{for $i = j$}
\end{cases}$$
Then $M = EF - FE$.
References

*

*K. Shoda (1936): Einige Sätze über Matrizen. Japanese J. Math. 13,
pp. 361--365.

*M. Rosset, S. Rosset (2000): Elements of trace zero that are not commutators.
Communications in Algebra 28:6, 3059-3071,
DOI: 10.1080/00927870008827009.

