The invention relates to a multi-component packaging unit (kit of parts) for oxidatively dyeing keratin fibers, comprising two preparations (A) and (B) packaged separate from each other, wherein (a) preparation (A) contains, in a cosmetic carrier, (a1) at least one oxidation dye pre-product and (a2) ammonia, and (b) preparation (B) contains, in a cosmetic carrier, (b1) hydrogen peroxide, (b2) at least one fatty alcohol from the group comprising arachidyl alcohol (eicosan-1-ol), gadoleyl alcohol ((9Z)-eicos-9-en-1-ol), arachidonic alcohol ((5Z, 8Z, 11Z, 14Z)-eicosa-5,8,11,14-tetraen-1-ol), heneicosyl alcohol (heneicosan-1-ol), behenyl alcohol (docosan-1-ol), erucyl alcohol ((13Z)-docos-13-en-1-ol), and/or brassidyl alcohol ((13E)-docosen-1-ol), and (b3) at least one hydrocarbon having 8 to 80 C atoms.
