## What is the first interesting matric Toda bracket in the stable homotopy of the sphere?

Feel free to gloss interesting' as you see fit. One way:

1. What is the lowest degree matric Toda bracket in $\pi_*(S)$ that doesn't contain zero?

By degree', I mean total homotopical degree, i.e. the $*$ in $\pi_*$. By matric' I mean to exclude ordinary Toda brackets, that is matric Toda brackets all of whose entries are one-by-one matrices, and also to exclude brackets that are trivially determined by ordinary Toda brackets.

I'm also interested in the title question with first' replaced by simplest'. For instance:

2. What is the lowest order matric Toda bracket in $\pi_*(S)$, all of whose matrix entries are sums of products of Hopf elements, that doesn't contain zero?

By order' I mean the number of entries in the bracket, i.e. whether it is 3-fold or 4-fold or 5-fold or ... .

Now I'd like to know the same, but with the proviso that the bracket be detectable in the classical Adams spectral sequence:

3. What is the first or simplest, interesting matric Toda bracket in $\pi_*(S)$ that is detectable in the $\mathrm{E}_2$ term of the classical Adams spectral sequence?

Some remarks:

• I believe the matric Massey product $\langle h_2^2, h_0, \left(\begin{array}{cc} h_1 h_3 & h_2^2 \end{array} \right), \left(\begin{array}{c} h_1 \\ h_2 \end{array}\right)\rangle$ in the $\mathrm{E}_2$ term of the Adams spectral sequence is the class $e_0$, but that is not a permanent cycle.
• Kochman lists the matric Toda bracket $\langle \sigma, \left(\begin{array}{cc} A[31] & \nu\end{array}\right),\left(\begin{array}{cc} \eta & 0 \\ 0 & \eta \end{array}\right),\left(\begin{array}{c} \nu \\ \eta A[30] \end{array}\right)\rangle$ in degree 44 as containing an element of order 8, where $A[30]$ and $A[31]$ refer to certain elements of order 2 in those degrees.
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(7.4) of arxiv.org/pdf/math/0311328v3.pdf ? – AndrĂ© Henriques Jun 20 at 15:01
Ah, so probably the bracket $\langle \begin{smallmatrix} \nu^2 & \eta \end{smallmatrix}, \begin{smallmatrix} \nu & \epsilon \\\ \epsilon & \nu \end{smallmatrix}, \begin{smallmatrix} \nu^2 & \eta \\\ \eta & \nu^2 \end{smallmatrix}, \begin{smallmatrix} \nu \\\ \epsilon \end{smallmatrix} \rangle$ is interesting, and detectable in Adams $\mathrm{E}_2$. – cdouglas Jun 20 at 15:51