(Note: This has been updated to be similar with this MO post.)
There are exactly 31 known primitive solutions to,
$$a^4+b^4+c^4 = d^4\tag{1}$$
with $d<10^{28}$. Noam Elkies showed that $(1)$ as,
$$(p + r)^4 + (p - r)^4 + s^4 = q^4\tag{2}$$
can be completely solved as an intersection of two quadric surfaces,
$$-(3 m^2 - 8m + 6) p^2 + 2 (m^2 - 2) p q - 2 m q^2 = (m^2 + 2) r^2\tag{3}$$
$$-4 (m^2 - 2) p^2 + 8 m p q + (m^2 - 2) q^2 = (m^2 + 2) s^2\tag{4}$$
for some constant $m$. Given a known solution to $(1)$, $m$ can be recovered as,
$$m = \frac{4p^2-q^2-s^2}{3p^2-2pq+r^2} = \frac{(a+b)^2-c^2-d^2}{a^2+ab+b^2-(a+b)d}\tag5$$
One can reverse-engineer the known solutions and find that there are only eight known rational $m$ of small height such that $(3),(4)$ can be rationally solved, namely,
$$m_k = -\frac{5}{8},\;-\frac{9}{20},\;-\frac{29}{12},\;-\frac{41}{36},\;-\frac{93}{80},\;-\frac{136}{133}$$
and positive,
$$m_k = \frac{201}{4},\;\frac{233}{60}$$
with the last one recently found by Andrew Bremner and yielding #21 mentioned in the comments below. The first three $m_k$ give rise to the conditional equations,
- $(-313+484v+85v^2)^4+(10-586v+68v^2)^4+(2t)^4=(363-204v+357v^2)^4$
- $(-15968 + 2334 v+59v^2)^4+(7068 + 3082 v + 10v^2)^4+(2t)^4 = (22628 + 54 v + 159v^2)^4$
- $(-11980 + 1673 v + 54v^2)^4+(36 - 2321 v + 3v^2)^4+(t)^4 = (24677 + 203 v + 71v^2 )^4$
with small solutions,
- $v = -31/467,\;\;-3015/9707,\;\;18247/19530,\;\;30671/229738$
- $v = 77/9,\;\;-1022/243,\;\; -50191/8685 $
- $v = -2020/127$
For example, factoring the first equation yields the elliptic curve condition,
$$22030 + 28849 v - 56158 v^2 + 36941 v^3 - 31790 v^4 = t^2$$
An initial point is $v=-31/467$ from which one can find an infinite more. These small points $v_i$ explain some of the 20 solutions with $d<10^{10}$, while the rest have unwieldy $m$. (The 3rd family might still have rational points that yield $d$ within that range.)
Question: What other $m$ is there of small height not in the list of eight above?
P.S: My thanks to Noam Elkies for help with a further family in the old version of this post.