# Good confusion-avoiding notation for iterated commutators?

I am doing some complicated and tedious calculation on iterated commutators. A typical term in my calculation looks like $$[x_a,[[[x_b,x_c]-x_d,x_e],[x_f,x_g]]]\text{.}$$

(I am considering commutators on Lie algebras so the minus sign $-$ is well defined here.) In the notation above there is no ambiguity; but when I have dozens of terms like that it looks horrible and a mistake in calculation is almost inevitable. I have simplified the notation a little bit by forgetting the $x$'s and write the commutator as $$[a,[[[b,c]-d,e],[f,g]]]\text{,}$$ but it still looks confusing. Not sure if I should continue to simplify it and simply write $$[a[[([bc]-d)e][fg]]]$$ instead. I tried to use physical papers as my draft papers and also opened a rich text document to note down the calculation; but both ways look awkward and it is difficult for me to "see" when should I cancel two terms (under some conditions) or use the Jacobi's identity.

So my question is:

Is there any "smart" notation that I can use for calculation like this and make the calculation less fallible?

I am not referring to writing for publishing, but just for calculation on one's own draft paper.

• I wonder if it would be difficult to try to get the computer to simplify expressions of this type. – MJD Dec 12 '14 at 15:55
• @MJD, are you referring to using some softwares such as GAP? – Zuriel Dec 12 '14 at 15:56
• How about $[x_a,x_b]=ab$? You'll still have to keep track of parentheses, but I think it uses about as little notation as possible. – user134824 Dec 12 '14 at 15:57
• I suggest to use larger brackets. $\big[a,[b,c]\big]$ – Martin Brandenburg Dec 12 '14 at 15:58
• @user134824, sounds good! I am also considering using different colours to keep track of parentheses; if I do so, I thinking using pens/pencils should be easier than using LaTex/Microsoft Word. – Zuriel Dec 12 '14 at 16:01

However, instead of $[X, Y]$, one could write, say, $\mathcal{L}_X(Y)$.
You could obviously even shorten this to $X(Y)$, like function notation. If you then switch to exponent notation for functions (as, $f(x) = x^f$), and especially if you're willing to mix these notations, then (assuming I parsed it right!) your sample formula could become, for instance: $$([f, g]^{[[b, c] - d, e]})^a$$ which to my eye is slightly more readable than what you had.
Obviously this could cause a bunch of other problems for you! For instance: (1) you may want to reserve exponential notation for, well, exponents. (2) Visually, it switched the order. So even though $a$ started out way on the left, now it's way on the right.
We could perhaps try to lessen the confusion in both (1) and (2) at once: Write ${}^xy = [x, y]$, so now your formula is $${}^a({}^{[[b, c] - d, e]}[f, g])$$ You could also use subscripts instead of superscripts, etc. It's a matter of taste whether something like this is preferable. Obviously, do whatever works best for you!
• Thank you for the suggestion! So you mean we write ${}^xy=[x,y]$ sometimes but not always, right? Since otherwise the notation will look like $${}^a({}^{^{{}^bc-d}e}({}^fg)).$$ And when to write ${}^xy$ and when to write $[x,y]$, there is no rule; right? – Zuriel Dec 12 '14 at 16:26
• Yeah, my suggestion would be to mix notations, assuming you're comfortable with that. So you could still use the $[x,y]$ notation sometimes. There wouldn't be have to be any rule for when to use each notation; you'd just use whatever you thought looked more clear. (Like how sometimes people write $2*3$ and sometimes people write $(2)(3)$; either is acceptable, and you just use whatever is most convenient at the time). – mathmandan Dec 12 '14 at 16:36