$B^2 = A$ for non diagonalizable $B \in \mathbb{C}^{4x4}$

Question

I have to prove that exist a non diagonalizable matrix $B \in \mathbb{C}^{4x4}$ such that $A = B^{2}$ with $ A= \begin{bmatrix} 1 & 2 & 6 & 0 \\ -1 & -2 & -6 & 0 \\ 0 & 1 & 4 & 0 \\ 2 & 4 & 12 & 0 \end{bmatrix} $

My problem is that A is diagonalizable with $ D= \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} $ and I can't imagine a Jordan matrix B satisfying $A = B^{2}$.

Answer 1

Hint: try looking for a block diagonal matrix of the form $$\begin{pmatrix} X & 0 \\ 0 & Y \end{pmatrix}$$ where $X, Y$ are $2 \times 2$ matrices, and $Y$ is non-zero and nilpotent.