One of my grey Cordings suits is my favourite - I am wearing it all year round, it is the perfect undercover shade of grey which always works.
I have it dry-cleaned twice a year, sometimes more often.
Tonight I have given the trousers a hand wash, with hair shampoo, in the washbasin.
The water was dark, I rinsed and repeated many times.
Curious to see if the shine will be gone after drying.
Were the subsequent rinses yielding clearer water? I'd be worried about colorfastness.
You could get away with it if the water was cold but you might find the cloth goes a bit floppy and shapeless.
Still if it makes you realise that shiny suits are not a good thing .... ever.......
Shine gone, and the fabric has become "stiffer", less greasy.
Now a navy trouser.
So a pair of navy trousers are next - or is the 'grey' suit now navy after a wash?
I've washed plenty of woolen tailored clothing, once even a reversible tweed/gabardine overcoat, over the years and it's generally been successful. I use the 'hand wash' setting on the machine with non-biological (i.e. for woolens) detergent, usually cold but if warm water is absolutely needed, no more than 30°. Air dry obviously.
In fact it's the same procedure for washing woolen or cashmere knitwear. Remember you can get wool either wet or warm - both at the same time lead to shrinkage.