A Lie algebra is a vector space (over a field; we'll usually use $\mathrm{\pi \x9d\x90\x91}$ or $\mathrm{\pi \x9d\x90\x82}$ ) $\mathrm{\pi \x9d\x94\u20ac}$ equipped with an operation $[,]:\mathrm{\pi \x9d\x94\u20ac}\Gamma \x97\mathrm{\pi \x9d\x94\u20ac}\beta \x86\x92\mathrm{\pi \x9d\x94\u20ac}$ satisfying:

bilinearity (over our chosen field $k$ ), that is the ability to expand brackets: $$[a\beta \x81\u2019X+b\beta \x81\u2019Y,Z]=a\beta \x81\u2019[X,Z]+b\beta \x81\u2019[Y,Z]\beta \x81\u2019\text{\Beta and\Beta}\beta \x81\u2019[X,a\beta \x81\u2019Y+b\beta \x81\u2019Z]=a\beta \x81\u2019[X,Y]+b\beta \x81\u2019[X,Z]\beta \x81\u2019\beta \x88\x80X,Y,Z\beta \x88\x88\mathrm{\pi \x9d\x94\u20ac},a,b\beta \x88\x88k$$

for all $X\beta \x88\x88\mathrm{\pi \x9d\x94\u20ac}$ we require $[X,X]=0$ . This makes sense if you think about the commutator bracket, because $[X,X]=X\beta \x81\u2019XX\beta \x81\u2019X=0$ .

the Jacobi identity: $$[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0\beta \x81\u2019\beta \x88\x80X,Y,Z\beta \x88\x88\mathrm{\pi \x9d\x94\u20ac}.$$ To remember this formula, note that the second and third terms are cyclic permutations of the first.