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To see this, I will include here some remarks on $\mathbb H$ and its automorphisms. Part of the OP's concern seems to be that it is not a priori automatic that metric concepts in $\mathbb H$ such as unit length or orthogonality (and hence the notion of being imaginary, since the imaginary quaternions are the orthogonal complement to $\mathbb R$ in $\mathbb H$) are preserved by Aut$(\mathbb H)$, and so one of my goals is to show that this concern is not necessary. Indeed, this geometry is intrinsic to the quaternions, as we will see. (This not coincidence: Hamilton was led to his discovery by trying to algebraize the geometry of $\mathbb R^3$.)

Now the source of this map can be identified with the quotient of the unit quaternions (which form a copy of $SU(2)$) by $\pm 1$, and of course $SU(2)/\{\pm 1\} = SO(3)$. On the other hand, this injection is in fact a bijection (i.e. any automorphism of $\mathbb H$ is inner), by the Skolem--Noether theorem. This puts the description of Aut$(\mathbb H^3)$ obtained above into a more general perspective.

To see this, I will include here some remarks on $\mathbb H$ and its automorphisms. Part of the OP's concern seems to be that it is not a priori automatic that metric concepts in $\mathbb H$ such as unit length or orthogonality (and hence the notion of being imaginary, since the imaginary quaternions are the orthogonal complement to $\mathbb R$ in $\mathbb H$) are preserved by Aut$(\mathbb H)$, and so one of my goals is to show that this concern is not necessary. Indeed, this geometry is intrinsic to the quaternions, as we will see. (This is not coincidence: Hamilton was led to his discovery by trying to algebraize the geometry of $\mathbb R^3$.)

Now the source of this map can be identified with the quotient of the unit quaternions (which form a copy of $SU(2)$) by $\pm 1$, and of course $SU(2)/\{\pm 1\} = SO(3)$. On the other hand, this injection is in fact a bijection (i.e. any automorphism of $\mathbb H$ is inner), by the Skolem--Noether theorem. This puts the description of Aut$(\mathbb H)$ obtained above into a more general perspective.

To see this, I will include here some remarks on $\mathbb H$ and its automorphisms. Part of the OP's concern seems to be that is not a priori automatic that metric concepts in $\mathbb H$ such as unit length or orthogonality (and hence the notion of being imaginary, since the imaginary quaternions are the orthogonal complement to $\mathbb R$ in $\mathbb H$) are preserved by Aut$(\mathbb H)$, and so one of my goals is to show that this concern is not necessary. Indeed, this geometry is intrinsic to the quaternions, as we will see. (This not coincidence: Hamilton was led to his discovery by trying to algebraize the geometry of $\mathbb R^3$.)

To see this, I will include here some remarks on $\mathbb H$ and its automorphisms. Part of the OP's concern seems to be that it is not a priori automatic that metric concepts in $\mathbb H$ such as unit length or orthogonality (and hence the notion of being imaginary, since the imaginary quaternions are the orthogonal complement to $\mathbb R$ in $\mathbb H$) are preserved by Aut$(\mathbb H)$, and so one of my goals is to show that this concern is not necessary. Indeed, this geometry is intrinsic to the quaternions, as we will see. (This not coincidence: Hamilton was led to his discovery by trying to algebraize the geometry of $\mathbb R^3$.)

To see this, I will include here some remarks on $\mathbb H$ and its automorphisms. Part of the OP's concern seems to be that is not {\em a priori} automatic that metric concepts in $\mathbb H$ such as unit length or orthogonality (and hence the notion of being imaginary, since the imaginary quaternions are the orthogonal complement to $\mathbb R$ in $\mathbb H$) are preserved by Aut$(\mathbb H)$, and so one of my goals is to show that this concern is not necessary. Indeed, this geometry is intrinsic to the quaternions, as we will see. (This not coincidence: Hamilton was led to his discovery by trying to algebraize the geometry of $\mathbb R^3$.)

To see this, I will include here some remarks on $\mathbb H$ and its automorphisms. Part of the OP's concern seems to be that is not a priori automatic that metric concepts in $\mathbb H$ such as unit length or orthogonality (and hence the notion of being imaginary, since the imaginary quaternions are the orthogonal complement to $\mathbb R$ in $\mathbb H$) are preserved by Aut$(\mathbb H)$, and so one of my goals is to show that this concern is not necessary. Indeed, this geometry is intrinsic to the quaternions, as we will see. (This not coincidence: Hamilton was led to his discovery by trying to algebraize the geometry of $\mathbb R^3$.)