Two theorems of Lie on infinitesimal symmetries of differential equations

We give an account, in terms of synthetic differential geometry, of some of Sophus Lie's geometric theory of first order differential equations.This theory is, in modern terms, formulated in terms of vector fields on manifolds. Some algebra of nilpotent elementsWe consider a commutative ring R. Recall that a R is nilpotent if a k = 0 for some natural number k = 1, 2, 3, . ... We are mainly concerned with the case k = 2, and defineThe letter 'd' will hence forward be reserved to elements of D.Note that commutativity of R implies that d D r d D, for any r R, in particular -d D. So the subset D is closed under multiplication.However, it is not closed under addition; indeed (. For x and y in R, we write x y if (x -y) 2 = 0.If R has the property that x + x = 0 implies x = 0 (which we henceforth assume), we therefore have 1.1.Proposition.For d 1 and d 2 in D,The n-dimensional analogue of D( 2) is D(n) D n , described by D(n) := {(d 1 . . ., d n ) R n | d i d j = 0 for all i, j = 1, . . ., n}.