Let $x \in \mathbb{R}$ and let $$P = \left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3\end{array}\right], \quad Q = \left[\begin{array}{ccc}2 & x & x \\ 0 & 4 & 0 \\ x & x & 6\end{array}\right] \text{ and } R = PQP^{-1}$$
Then which of the following options is/are correct?
(A) There exists a real number $x$ such that $PQ = QP$
(B) $\det R = \det\left[\begin{array}{lll}2 & x & x \\ 0 & 4 & 0 \\ x & x & 5\end{array}\right] + 8$, for all $x \in \mathbb{R}$
(C) For $x = 0$, if $R\left[\begin{array}{l}1 \\ a \\ b\end{array}\right] = 6\left[\begin{array}{l}1 \\ a \\ b\end{array}\right]$, then $a + b = 5$
(D) For $x = 1$, there exists a unit vector $\alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$ for which $R\left[\begin{array}{l}\alpha \\ \beta \\ \gamma\end{array}\right] = \left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$