We know that the real 3-dimensional sphere is isomorphic and diffeomorphic to various Lie groups $$S^3 = Spin(3) = SU(2).$$
While the larger dimensional $Spin$ group has a close relation to $Spin(3)$, since $ Spin(m) \supset Spin(3) $
My question is that when $m=2k$, whether we can "visualize" the geometry of $ Spin(2k) $ via the way that various many $S^3$ are embedded inside $Spin(2k)$? Also the way that how each $S^3$ can move around to the other $S^3$ inside $Spin(2k)$?
$$ Spin(2k) \supset Spin(3). $$
Naively, we need 3 generators of Lie algebra out of the $k(2k-1)$ generators to span the $S^3=Spin(3)$.
When $k=2$, we get the $$Spin(4)=Spin(3) \times Spin(3)=S^3 \times S^3 \supset Spin(3) = S^3.$$ This is simply a product of two $S^3$, easy to visualize.
How about larger $Spin(6)=SU(4),Spin(8),etc.$ Can we visualize them via many $S^3$ spheres embedded inside?
This is the end of my question. You can use your own imagination to come up a visualization. I also provide some of my thoughts below.
Here is one approach of mine:
For example, it is known that the $Spin(2k)$ is a real $k(2k-1)$-dimensional connected and simply connected manifold. It can be generated by the $2^{k}$-dimensional reducible spin representation of Lie algebra with a generator defined by the commutator $[\gamma^a,\gamma^b]=\gamma^a \gamma^b- \gamma^b \gamma^a$, as $$ M^{ab}=\frac{i}{4}[\gamma^a,\gamma^b]=M^{ba} $$ with $i$ is the imaginary $i:=\sqrt{-1}$. Here each $\gamma^a$ or $\gamma^b$ (with $a,b=1,2,\dots,2k$) is a rank-$2^k$ matrix (namely, a $2^k$ by $2^k$ dimensional matrix).
With the standard rank-2 Pauli matrices: $$ \sigma_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \quad \sigma_2 = \begin{pmatrix} 0& -i \\ i&0 \end{pmatrix} \quad \sigma_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}. $$
We can obtain the $2^{k-1}$-dimensional reducible spin representation of Lie algebra, if we do a projection $$ \frac{1 + \gamma^{2k+1}}{2} $$ or a projection $$ \frac{1 - \gamma^{2k+1}}{2}. $$ Then the full Lie group shall be able to be spanned via the Lie algebra-Lie group correspondence writing the exponential of the $k(2k-1)$ generators of $M^{ij}$: $$ \exp(\dots i \theta_{ij} M^{ij}). $$
Question: when $m=2k$, whether we can "visualize" the geometry of $ Spin(2k) $ via the way that the $S^3$ are embedded inside $Spin(2k)$?