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Define morphism of Q[i,w] in terms of i and w

I want to construct the field $K=Q[i,w]$ where $w=\sqrt[4]{2}$ and define a field homomorphism such as $w\mapsto iw$, $i\mapsto -i$. I'd like to do

K.<w,I> = NumberField([x^4-2,x^2+1])
H = End(K)

but it doesn't work because $K$ is considered to be the relative field $(Q[i])[w]$. I know I could work with the absolute field

K_abs.<theta> = K.absolute_field()

but I'd really like to define the morphism in terms of $w,i$. Is it possible?