Let \(G\) be a simply-connected simple Chevalley group of type \(A\), \(D\) or \(E\) and \(k\) an algebraically closed field whose characteristic is very good for \(G\). According to a conjecture of Grothendieck a semi-universal deformation (also known as miniversal deformation) of the rational double point of the same type can be obtained via a restriction of the categorical quotient morphism \(G\to G//G_{ad}\). More precisely, one has to restrict this morphism to a general slice through a point in the subregular unipotent orbit. In characteristic zero this was proved by \textit{E. Brieskorn} [Actes Congr. internat. Math. 1970, 2, 279-284 (1971; Zbl 0223.22012)] and \textit{P. Slodowy} [Lect. Notes Math. 815 (1980; Zbl 0441.14002)]. Slodowy extended this result to the case where \(\text{char}(k)>4 \operatorname {Cox}(G)-2\). Using work of \textit{V. Hinich} [Isr. J. Math. 76, No. 1/2, 153-160 (1991; Zbl 0810.14001)], this article gives a complete proof of the conjecture, i.e.~there is no assumption on \(\text{char}(k)\) except that it must be very good for \(G\). (\(p\) is very good for \(E_8\) if \(p\neq 2,3,5\); for \(E_6\) and \(E_7\) if \(p\neq 2,3\); for \(D\) if \(p\neq 2\) and for \(A_r\) if \(p\) does not divide \(r+1\).).