Neat from the bottle on my natural white bangs ....
30 mins later Washed out and failed
No purple hue just a lack of white bangs ... they just look grey like the rest.
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Something like a lavender needs a very light banana peel like color or fully white to take. Your base still looks a bit too grey imo for the lavender to have taken correctly. I think you might be better off taking a higher pigmented purple like SFX deep purple and diluting it w white conditioner to take over better what your current color is.
"If you don't know where you're going, any road will get you there." <---Lewis Carrol
You said your hair is 'natural white', is that from greying of your hair? , because naturally grey hair can be very dye resistant, it's hard to get any dye to stick. Trying to soften it first with a peroxide-only soak has helped other people. Their are some older forum posts on it.