the fact that in ${\mathbb P}(E({\lambda}))$ there is only one point fixed by the diagonal action of $G$ (the line of scalar matrices) proves that any $G\times G$ linear projective compactification of $G_{\mathrm{ad}}$ is of the form $X_\Sigma$. A projective $G\times G$-variety $X$ is said to be *linear* if there exists an equivariant embedding $X\subset {\mathbb P}(V)$ where $V$ is a finite dimensional rational $G\times G$-module. In particular as a consequence of Sumihiro’s Theorem (see for example [@KKLV Corollary 2.6]) all normal projective compactifications are linear. In this paper we study only linear compactifications.

Normality {#sez:normalita}
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In this section we determine for which simple $\Sigma$ the variety $X_\Sigma$ is normal, proving in particular Theorem A. In the following, by $\lambda$ we will always denote the maximal element of $\Sigma$.

Let ${\varphi}_{\lambda}\in {E({{\lambda}})}^*$ be a highest weight vector and set $X_\Sigma^\circ \subset X_\Sigma$ the open affine subset defined by the non-vanishing of ${\varphi}_{\lambda}$. In particular, we set ${\widetilde}X_{\lambda}=
X_{{\Pi^+({{\lambda}})}}$ and notice that ${\widetilde}X ^\circ_{\lambda}=
r^{-1}(X_\Sigma^\circ)$. Notice that $X^\circ_\Sigma$ is $B\times B$-stable and, since it intersects the closed orbit, it intersects every orbit: therefore $X_\Sigma$ is normal if and only if $X_\Sigma^\circ$ is normal if and only if the restriction $r{\bigr|}_{{\widetilde}X
