Radical of a quotient Lie algebra
Solution 1:
To simplify the notation write $R$ for ${\rm Rad}$ and let the Lie algebra be $L$ and the ideal $I$. Put $X/I=R(L/I)$. Then $R(L)\cap I=R(I)$, so we have $(R(L)+I)/I\cong R(L)/R(I)$. Clearly $R(L)+I\subseteq X$, so, if $L=R(L)\dot{+} S$ is the Levi decomposition of $L$, $X=R(L)+ X\cap S$. But $X\cap S$ is a semisimple ideal of $S$, and so $X\cap S= (X\cap S)^{(n)} \subseteq I$ for some $n \in \mathbb{N}$. Hence $X \subseteq R(L)+I$ and $R(L)/R(I)\cong (R(L)+I)/I= X/I=R(L/I)$.