Taylor Algebra?

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Some guy I met in the library today was explaining some mathematics to me and he mentioned what I think he called a "Taylor algebra". I later looked it up online and couldn't find anything by that name. Does any one know what it is? Here are some rules he wrote down in my note book:

$$[a,b+c]_n = [[a,b]_n,c]_n = [[a,c]_n,b]_n$$

$$[a+b,c]_n = [[a,c]_n,b]_0 = [[b,c]_n,a]_0$$

$$[a,b]_n + [b,a]_n = [a+b,a+b]_n = [a,a]_n + [b,b]_n$$

$$[a,[b,c]_n]_m = [[a,b]_m,c]_{n+m} = [[a,c]_{m+n}, b]_m$$

$$[[a,b]_m,c]_n = [a,[b,c]_{n-m}]_m = [a, [c,b]_{m-n}]_n$$​

I didn't really understand what he was saying and I don't know if this is a definition of the algebra or just some properties.

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"Taylor Algebra?"

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