I'm looking for a source of properties for semisimple Lie algebra elements, specifically finite dimensional classical Lie algebras.

I start with the assumption that I have a complexified Lie algebra $\mathfrak{g}^\mathbb{C}$, with a semisimple element $\Lambda \in \mathfrak{g}^\mathbb{C}$, and that it's conjugate $\bar{\Lambda}$ (under the complex structure of $\mathfrak{g}^\mathbb{C}$) has the property that, $$\left[ \Lambda , \bar{\Lambda}\right] = 0$$ Does this imply that $ker\> ad \> \Lambda = ker\> ad \> \bar{\Lambda}$, I know that in many cases I've worked with, it does, yet is there a reason why in general this is or isn't true?