there is nothing interesting about your second question. it is very straightforward. so i'll answer your first question for now and i'll help you with the second later if you tried and still couldn't do it.

for any lie algebra

let

and inductively define

and

an easy induction shows that

. thus if

is nilpotent, then

is solvable.

conversely, suppose that

is solvable. then, by Lie's theorem,

has a basis

such that

the matrix of

with respect to

is upper triangular for every

it is easy to see that if

and

are two upper triangular matrices, then

is strictly upper triangular and hence nilpotent. thus

is nilpotent. so we have proved that

is nilpotent for all

and therefore, by Engel's theorem,

is nilpotent.