A wool garment's label determines whether it belongs in a machine wash, a hand wash or neither. Woolmark's symbol guide explains that a very mild washing process reduces mechanical treatment and may require a smaller load. That is a different instruction from assuming any gentle-looking setting is suitable for every wool item.

Drying and bleaching instructions must also be read separately. The guide distinguishes do not tumble dry from a permitted mild drying process, and identifies the crossed triangle as a prohibition on bleach. Check the whole set of care directions rather than using the washing temperature as permission for every later step.
GarmentCare Guide 