Can you give more detail about what sort of answer you're seeking? The groups arising in this way are precisely those that are anisotropic modulo centre—i.e., for which the only split tori are central. (Indeed, the construction just returns the original group if one starts with such a group.) I may have the terminology wrong, but I think that the centraliser arising in this way (which is unique up to conjugacy) is called the anisotropic kernel of the original group. Indeed, as Jim mentions, Borel and Tits have developed a huge amount of structural information for these sorts of groups; in addition to Groupes réductifs, for the case of groups over local fields I recommend Tits's Corvallis paper. There are some surprises: For example, in the $p$-adic case, the only anisotropic groups are forms of $A_n$.

A truly trivial observation (because it's essentially the definition) is that you wind up with an Abelian centraliser (equivalently, a torus) if and only if your original group was $k$-quasi-split.

UPDATE: Having just read the linked question, I see that you were requesting examples. I tend to think of one of the 2 extremes: either something like the quasi-split $\operatorname{SO}(3)$, where we get a torus, or an already-anisotropic-modulo-centre group like $\operatorname{GL}_1(D)$. Then again, all my intuition is in the $p$-adics, so, as I mentioned, there simply aren't that many interesting kinds of non-Abelian-ness that can occur.

Can you give more detail about what sort of answer you're seeking? The groups arising in this way are precisely those that are anisotropic modulo centre—i.e., for which the only split tori are central. (Indeed, the construction just returns the original group if one starts with such a group.) I may have the terminology wrong, but I think that the centraliser arising in this way (which is unique up to conjugacy) is called the anisotropic kernel of the original group. Indeed, as Jim mentions, Borel and Tits have developed a huge amount of structural information for these sorts of groups; in addition to Groupes réductifs, for the case of groups over local fields I recommend Tits's Corvallis paper. There are some surprises: For example, in the $p$-adic case, the only anisotropic groups are forms of $A_n$.

A truly trivial observation (because it's essentially the definition) is that you wind up with an Abelian centraliser (equivalently, a torus) if and only if your original group was $k$-quasi-split.

UPDATE: Having just read the linked question, I see that you were requesting examples. I tend to think of one of the 2 extremes: either something like the quasi-split $\operatorname{SO}(3)$, where we get a torus, or an already-anisotropic-modulo-centre group like $\operatorname{GL}_1(D)$. Then again, all my intuition is in the $p$-adics, so, as I mentioned, there simply aren't that many interesting kinds of non-Abelian-ness that can occur.

Can you give more detail about what sort of answer you're seeking? The groups arising in this way are precisely those that are anisotropic modulo centre—i.e., for which the only split tori are central. (Indeed, the construction just returns the original group if one starts with such a group.) I may have the terminology wrong, but I think that the centraliser arising in this way (which is unique up to conjugacy) is called the anisotropic kernel of the original group. Indeed, as Jim mentions, Borel and Tits have developed a huge amount of structural information for these sorts of groups; in addition to Groupes réductifs, for the case of groups over local fields I recommend Tits's Corvallis paper. There are some surprises: For example, in the $p$-adic case, the only anisotropic groups are forms of $A_n$.

A truly trivial observation (because it's essentially the definition) is that you wind up with an Abelian centraliser (equivalently, a torus) if and only if your original group was $k$-quasi-split.

UPDATE: Having just read the linked question, perhaps you were requesting examples? I tend to think of one of the 2 extremes: either something like the quasi-split $\operatorname{SO}(3)$, where we get a torus, or an already-anisotropic-modulo-centre group like $\operatorname{GL}_1(D)$. Then again, all my intuition is in the $p$-adics, so, as I mentioned, there simply aren't that many interesting kinds of non-Abelian-ness that can occur.

Can you give more detail about what sort of answer you're seeking? The groups arising in this way are precisely those that are anisotropic modulo centre—i.e., for which the only split tori are central. (Indeed, the construction just returns the original group if one starts with such a group.) I may have the terminology wrong, but I think that the centraliser arising in this way (which is unique up to conjugacy) is called the anisotropic kernel of the original group. Indeed, as Jim mentions, Borel and Tits have developed a huge amount of structural information for these sorts of groups; in addition to Groupes réductifs, for the case of groups over local fields I recommend Tits's Corvallis paper. There are some surprises: For example, in the $p$-adic case, the only anisotropic groups are forms of $A_n$.

A truly trivial observation (because it's essentially the definition) is that you wind up with an Abelian centraliser (equivalently, a torus) if and only if your original group was $k$-quasi-split.

UPDATE: Having just read the linked question, I see that you were requesting examples. I tend to think of one of the 2 extremes: either something like the quasi-split $\operatorname{SO}(3)$, where we get a torus, or an already-anisotropic-modulo-centre group like $\operatorname{GL}_1(D)$. Then again, all my intuition is in the $p$-adics, so, as I mentioned, there simply aren't that many interesting kinds of non-Abelian-ness that can occur.