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It is commonly known that we have a chain of embeddings

$$SU(4)\subset Spin(7)\subset SO(8)$$

(there is more than one possible $Spin(7)$, just take one).

Which is the explicit analog for the Lie Algebras embeddings? How may we describe the 21-dimensional space corresponding to $\mathfrak{so}(7)$ in

$$\mathfrak{su}(4)\subset \mathfrak{so}(7)\subset \mathfrak{so}(8)$$

according to their matrices?

And also,

How to characterise the image of a Cartan subalgebra of $\mathfrak{so}(7)$?

Any suggestion is welcome.

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    $\begingroup$ You'll find all of this and more in F. R. Harvey's book "Spinors and Calibrations". $\endgroup$
    – Robert Bryant
    Commented Jun 10, 2015 at 11:39
  • $\begingroup$ @RobertBryant Thank you very much!! Which chapter / page approximately? $\endgroup$
    – Jjm
    Commented Jun 11, 2015 at 6:31
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    $\begingroup$ Look at the material in Chapter 14 in the section "Triality", where $\mathrm{Spin}(7)\subset\mathrm{SO}(8)$ is described as the group generated by $R_u$ for $u\in S^6\subset\mathrm{Im}(\mathbb{O})$. (He's identifying $\mathbb{R}^8$ with $\mathbb{O}$, the octonions.) While the Lie algebra isn't written out explicitly there, it follows immediately from Harvey's Lemma 14.66 that ${\frak{spin}}(7)\subset{\frak{so}}(8)$ is the space of skew-symmetric matrices orthogonal to all the skew-symmetric matrices $R_u$ for $u\in\mathrm{Im}(\mathbb{O})$. $\endgroup$
    – Robert Bryant
    Commented Jun 11, 2015 at 11:42
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    $\begingroup$ Oh, also: Any Cartan subalgebra of ${\frak{su}}(4)$ is also a Cartan subalgebra of ${\frak{spin}}(7)$, since both these groups have rank $3$. $\endgroup$
    – Robert Bryant
    Commented Jun 11, 2015 at 11:49
  • $\begingroup$ @RobertBryant Thank you...! Your answers are simply brilliant, they always help me a lot. $\endgroup$
    – Jjm
    Commented Jun 11, 2015 at 12:19

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