What does "antisymmetric" mean for the adjoint map of a Lie algebra?

Question

A theorem about Lie algebras says that:

Fix a Lie group $G$, endowed with a left invariant riemannian metric $g$, let $\nabla$ be the Levi Civita connection. Let the Lie algebra of $G$ be $\mathfrak{g}$ then the following are equivalent:

1. The adjoint map of $\mathfrak{g}$ is such that $ad(X)$ is antisymmetric $\forall X\in\mathfrak{g}$
2. The one parameter subgroups of $G$ are precisely the (Levi-Civita) geodetics of $G$

Now my question is not about the theorem, but about the meaniing of the first condition. To my understanding $ad$ is a linear map $X\in\mathfrak{g}\mapsto ad(\mathfrak{g})\in \mathfrak{gl}(\mathfrak{g})\cong Hom(\mathfrak{g},\mathfrak{g})$ where $ad(X): Y\in\mathfrak{g}\mapsto [X,Y]\in \mathfrak{g}$. So it is a linear map, and not multinear map: how can it be antisymmetric? Wat do we mean by this term here? What am I missing?

Thanks in advance

Answer 1

I believe there exists an underlying differentiable left invariant scalar product, antisymmetric means $\langle ad_x(y),z\rangle+\langle y,ad_x(z)\rangle=0$.

If they do not talk about the metric and the group is semi-simple take the Killing form.