# FedCoin: A Peer-to-Peer Payment System for Federated Learning

Yuan Liu1111Contact Author Shuai Sun1 Zhengpeng Ai1&Shuangfeng Zhang1 Zelei
Liu2 Han Yu2
1Northeastern University, Shenyang, China
2Nanyang Technological University, Singapore
<EMAIL_ADDRESS>sunshuai.edu@gmail, <EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Federated learning (FL) is an emerging collaborative machine learning method
to train models on distributed datasets with privacy concerns. To properly
incentivize data owners to contribute their efforts, Shapley Value (SV) is
often adopted to fairly assess their contribution. However, the calculation of
SV is time-consuming and computationally costly. In this paper, we propose
FedCoin, a blockchain-based peer-to-peer payment system for FL to enable a
feasible SV based profit distribution. In FedCoin, blockchain consensus
entities calculate SVs and a new block is created based on the proof of
Shapley (PoSap) protocol. It is in contrast to the popular BitCoin network
where consensus entities “mine” new blocks by solving meaningless puzzles.
Based on the computed SVs, a scheme for dividing the incentive payoffs among
FL clients with nonrepudiation and tamper-resistance properties is proposed.
Experimental results based on real-world data show that FedCoin can promote
high-quality data from FL clients through accurately computing SVs with an
upper bound on the computational resources required for reaching consensus. It
opens opportunities for non-data owners to play a role in FL.

##  1 Introduction

Nowadays, many businesses generate large amount of data through usage, and
rely on making informed decisions using machine learning (ML) based on such
data in order to thrive. With changing regulatory scene, ML is facing
increasingly difficult challenges with respect to the usage of such data. Data
are often collected and owned by multiple distributed entities. They often
contain sensitive or private information and is cannot be stored in a
centralized server without violating privacy protection laws. In recent years,
federated learning (FL) has emerged as a promising solution to these
challenges Yang et al. (2019a).

In FL, each entity trains its local model and contributes the local model
parameter updates to a center server to help build a more powerful global FL
model. Compared with centralized ML methods, FL not only reduces communication
costs by transmitting model updates instead of raw data, but also reduce the
computational costs of the server by leveraging computing power from the
clients. Moreover, since local data never leave the data owners, FL improves
user privacy McMahan et al. (2017); Bonawitz et al. (2019).

From the above description, it is clear that FL clients are making significant
contributions towards the FL model. Thus, in order to sustain an FL community,
it is important for FL clients to be properly incentivized. For this to
happen, FL clients must be treated fairly Yu et al. (2020). Existing FL
incentive schemes generally agree that fair treatment of FL clients shall be
based on a fair assessment of their contributions to the FL model Kairouz et
al. (2019). Currently, the most widely adopted method for fair assessment of
FL client contribution is that of Shapley Values (SVs) Jia et al. (2019); Song
et al. (2019).

SV is a popular notion in fairly distributing profits earned by a coalition
among its contributors. It has been applied in various fields, ranging from
economics, information theory, and ML. The reason for its broad application is
that the SV divides the payoff with attractive properties such as fairness,
individual rationality, and additivity. However, SV based distribution
solution often takes exponential time to compute with a complexity of
$\mathcal{O}(n!)$ where $n$ is the number of data items. Even though the
computational complexity can be reduced through approximating SV with marginal
error guarantees Jia et al. (2019), it is still computationally costly.

In order to help FL systems compute SVs to support sustainable incentive
schemes, we propose a blockchain-based peer-to-peer payment system FedCoin.
The Shapley value of each FL client, reflecting its contribution to the global
FL model in a fair way, is calculated by the Proof of Shapley (PoSap)
consensus protocol which replaces the traditional hash-based protocol in
existing proof of work(PoW) based blockchain systems. All the payments are
recorded in the block in an immutable manner. Under FedCoin, paying out
incentives to FL clients does not need to rely on a central FL server. Based
on this, FedCoin provides a decentralized payment scheme for FL so that
incentives for all participants can be delivered in third-party-free manner
with nonrepudiation and tamper-resistance properties.

Extensive experiments based on real-world data show that FedCoin is able to
properly determine FL clients’ Shapley Value-based contribution to the global
FL model with an upper bound on the computational resources required for
reaching consensus. To the best of our knowledge, FedCoin is the first attempt
to leverage blockchain technology in federated learning incentive scheme
research. It opens up new opportunities for entities which has computational
resources but without local data to contribute to federated learning.

##  2 Related Work

The incentive mechanism design is an important research direction in the field
of federated learning Kairouz et al. (2019); Yang et al. (2019a). In Kang et
al. (2019), the contract theory is employed to improve the accuracy of model
training considering the unreliable data contributors. A consortium blockchain
architecture is applied to build a decentralized reputation system. In Khan et
al. (2019), a Stackelberg-game based incentive mechanism is designed to
optimize the utilities of both FL clients and the FL server. These works focus
on optimizing the rewards for self-interested FL clients and FL customers who
pay to use the FL model. Our study is compatible with these works in terms of
determining the payment budget for the FL customers.

In field of ML, SV has also be studied widely for various purpose. SV can be
applied in feature selection, ranking the importance of training data, which
is further applied in explaining the behavior of ML models Li and Cui (2019).
Since the computation complexity is $\mathcal{O}(n!)$, approximations of SV
also attract many attentions. In Ancona et al. (2019), a polynomial-time
approximation of SV is proposed for deep neural network. Group sampling based
approximation is studied in Jia et al. (2019). In this work, our objective is
not to decrease the computational complexity, but to establish a scheme so
that distributed computational resources, which are otherwise wasted, can be
leveraged to help FL systems calculate SVs.

Blockchain has been widely applied in addressing the security problems in FL
applications Dillenberger et al. (2019); Kairouz et al. (2019); Yang et al.
(2019a). FLChain Bao et al. (2019) and BlockFL Kim et al. (2019) have been
proposed to record the local model parameter updates in the temper-resistant
manner. A blockchain-based FL was proposed in Ramanan et al. (2019) so as to
remove the need for an FL server. A blockchain-based trust management system
was proposed in Kang et al. (2019) to assist FL server to select reliable and
high quality data owners as FL clients. These blockchain systems are used as
immutable ledgers to record local gradients and aggregate them in a trusted
manner. Our work will adopt the blockchain network as a computational engine
and payment distribution ledger, which is the first of its kind in the current
literature.

##  3 Preliminaries

For a typical FL scenario, we take
$\mathcal{F}_{i}(w)=\ell(x_{i},y_{i};w_{t})$ as the loss of prediction on a
sample ($x_{i}$,$y_{i}$) with model parameters $w$ at the $t$-th round. The
parameters $w^{t}$ is a $d$-dimensional vector. We assume that there are $K$
clients, and each client has a local data set $\mathcal{D}_{k}$ with
$n_{k}=|\mathcal{D}_{k}|$. The overall dataset is
$\mathcal{D}=\\{\mathcal{D}_{1},\ldots,\mathcal{D}_{K}\\}$ with
$n=|\mathcal{D}|=\sum_{k=1}^{K}n_{k}$. The objective function to be optimized
is:

$\min_{w\in\mathcal{R}^{d}}\mathcal{F}(w)\text{\hskip 14.22636ptwhere \hskip
14.22636pt}\mathcal{F}(w)=\frac{1}{n}\sum_{k=1}^{K}\sum_{i\in\mathcal{D}_{k}}\mathcal{F}_{i}(w)$
(1)

This optimization problem is generally solved by stochastic gradient descent
(SGD) Goodfellow et al. (2016) based methods. For example, based on the
current model $w_{t}$, the federated averaging algorithm McMahan et al. (2017)
computes the average gradient
$g_{k}^{t}=\frac{1}{n_{k}}\sum_{i\in\mathcal{D}_{k}}\nabla\mathcal{F}_{i}(w^{t})$
on the local data of client $k$. Each client updates its local model
$w_{k}^{t+1}\leftarrow w_{t}-\eta g_{k}^{t}$, and the FL server aggregates the
local models as the global FL model.

$w_{t+1}\leftarrow\mathcal{A}(\\{w_{t+1}^{k}|k=1,\ldots,K\\})$ (2)

where $\mathcal{A}$ is an aggregation function.

##  4 FedCoin

There are two networks of participants in our system: 1) the FL network, and
2) the peer-to-peer blockchain network (Figure 1).

Figure 1: Overview of the Proposed Model

A FL model requester or FL training task requester refers to the entities who
need to train an FL network and with a budget of $V$. In the FL network, there
is a centralized server, referred as FL server, in coordinating the executing
of model training and receiving payment $V$ from FL model requester.

The distributed data owners, called as FL clients, participate in a
collaborative training task and receive a payment $V$. Each FL client trains
its local model and submits the parameter updates to the FL server. The FL
server plays three roles. Firstly, it publishes a training task to FL clients
with price TrainPrice. Secondly, it aggregates local updates through a secure
aggregation protocol Bonawitz et al. (2017) and earns a computation payment
(ComPrice). Thirdly, it transfer a processing fee SapPrice to the blockchain
network to enlist its members’ help in calculating the FL model. The total
payment of the task (TrainPrice+ComPrice+SapPrice) should be not greater than
$V$ in order to sustain payment balance without relying on external transfer
of values into this system.

After each global model update, the FL server publishes a task to calculate
the contribution by each FL client. The consensus nodes in blockchain network
then collaboratively calculate SVs, and the block winner receives a payment of
TrainPrice+SapPrice. The winner then divides ComPrice to FL clients according
to their respective SVs by creating transactions in the blockchain. In our
current design, we only reward clients with postive contributions, but refrain
from penalizing clients with negative contributions. All the transactions are
recorded in the new block and further updated to the chain.

Therefore, the connection between FL network and blockchain network is a
special type of task. A task includes the received local update set
$W=\\{w_{k}|k=1,\ldots,K\\}$, the aggregation function $\mathcal{A}$, the loss
function $\mathcal{F}(w)$, and values SapPrice and TrainPrice for each update
round. Note that SapPrice and TrainPrice decreases as the number of training
rounds increases, and the total payment for training can be divided among the
rounds equally or not. Without loss of generality, the following description
focuses on a single training round.

###  4.1 Shapley Value Based Blockchain Consensus

Upon receiving a Shapley value calculation task from the FL network, the
miners in the blockchain network are to calculate the SV vector
$S=[s_{k}]_{k\in[1,K]}$ where $s_{k}$ is the SV of the client in providing
$w_{k}\in W$. Each miner independently calculates the SV vector following
Algorithm 1. Since the objective of the mining process is to competitively
calculate SV vectors so as to prove the miner’s computation power, we name the
algorithm as “Proof of Shapley (PoSap)”. The input of Algorithm 1 comes from
the task specifications from the FL network. The output is a new generated
block.

Input: $\mathcal{F}$: Loss function;

$\mathcal{A}$: Aggregation function;

$W$: Contribution of FL clients in size $K$;

$D$: Difficulty in Mining;

Output: $Blk$: a new block

1 Initialize $S=[s_{k}=0|k=1,\ldots,K]$;

2 time=0;

3 while  _No received $Blk$ OR !VerifyBlock($Blk$)_ do

4 $S_{t}=[s_{k}=0|k=1,\ldots,K]$% temporary store $S$

5 Random generate a rank $R=[r_{k}|k=1,\ldots,K]$;

6 $S_{t}(R(1))=\mathcal{F}(\mathcal{A}(W(R(1))))$;

7 for  _$i$ from $2$ to $K$_ do

8 $S_{t}(R(i))=\mathcal{F}(\mathcal{A}(W(R(1:i))))$;

9 $S_{t}(R(i))=S_{t}(R(i))-\sum_{j=1}^{i-1}S_{t}(R(j))$;

10

11 end for

12 $S=\frac{S\times time+S_{t}}{time+1}$;

13 time=time+1;

14 Broadcast $S$ and time;

15

16 end while

17if  _Receive a new $S$_ then

18 Average the Received $S$ to $\overline{S}=\frac{\sum time\times S}{\sum
time}$;

19 if  _$\|S-\overline{S}\|_{p}\leq D$_ then

20 Create a new block $Blk$ after longest chain;

21 Broadcast $Blk$;

22 return $Blk$;

23 end if

24

25 end if

26if  _Receive a new $Blk$_ then

27 if  _VerifyBlock( $Blk$)==ture_ then

28 Update $Blk$ to its chain;

29 return $Blk$;

30 end if

31

32 end if

Algorithm 1 Proof of Shapley (PoSap)

In Algorithm 1, a miner first initializes the SV vector as an all-zero vector,
and sets the calculation iteration number as $0$ (Line 1-2). The SV
computation continues as long as one of the following two conditions are
satisfied: 1) there is no new block received; or 2) the received block fails
to pass the verification which is specified in Algorithm 2 (Line 3). The SV
calculation process is described in Line 4 to Line 13. The miner initializes a
temporary SV vector $S_{t}$ to record the calculated value in this iteration
(Line 4). Then, the miner randomly generates a rank of the $K$ FL clients
(Line 5). According to the rank, an SV of the first entity is calculated as in
Line 6, which is the contribution of the entity to the loss function (Line
6)). For the next entity $i$, the Shapley value is calculated as its marginal
contribution (Line 7-10). $S$ is updated by averaging all the previous
iterations and the current $S_{t}$ (Line 11). The iteration time is then
incremented by 1 (Line 12). Then, the entity broadcast $S$ and time (Line 13).

Whenever a miner receives $S$ and $time$, the miner calculates the average
results $\overline{S}$ of all the received $S$ (Line 16). Then, the miner
calculates the $P$-distance between its own $S$ and $\overline{S}$. When the
distance is no greater than the mining difficulty $D$, the miner becomes the
winner and generates a new block $Blk$ (Line 18). The difficulty $D$ is
adapted dynamically as explained in Section 4.2. The illustration of the
Shapley based verification is shown in Figure 2. The new block is then
appended to the current longest chain.

Figure 2: Shapley Valued based Consensus Protocol

Whenever a miner receives a new block $Blk$, the miner verifies this block
according to Algorithm 2. Once the verification passes, the block is appended
to the miner’s chain, and the mining process terminates (Line 23-28).

The structure of a block is shown in Figure 3, including block header and
block body. The block header includes seven pieces of information, Table 1
presents the explanation about each header item. The block body records two
types of data: (1) The task specification including all the inputs for
Algorithm 1; and (2) The transactions in the blockchain network. Here, a
transaction is denoted as a certain amount of FedCoins transferred from a user
account to another, which is similar to that in BitCoin Nakamoto and others
(2008). The block winner has the privilege to create special transactions:
transferring TrainPrice from its own account to to the FL clients according to
$\overline{S}$. The detailed design is explained in Section 4.3.

Figure 3: Block Structure in FedCoin Table 1: Explanation of Block Header Type |  Explanations
---|---
Block ID |  The block height
Winner’s ID |  Identity of the block generator
Averaged $S$ |  The calculated Shapley vector in Line 15 of Algorithm.1
Previous Hash |  Hash value of previous block based on a hash function, e.g. SHA 256
Winner’s $S$ |  The Shapley vector calculated by the winner
Difficulty |  The required difficulty $D$
Merkel Tree Root |  Root of Merkel tree organized with transactions in block body

The verification procedure is described as in Algorithm 2. Three conditions
must be satisfied for a block to successfully pass the verification. The first
condition is $\|S_{t}-\overline{S}_{t}\|_{p}\leq D$ which aims to verify
whether the winner has generated the block with a valid SV calculation result.
The second condition is $\|\overline{S}-\overline{S}_{t}\|_{p}\leq D$ which
requires that the $\overline{S}$ value of the block should be close enough to
the local aggregated $S$. $\overline{S}$ should be equal to $\overline{S}_{t}$
when the blockchain network is synchronized. In an asychronized network, this
condition requires that the winner should aggregate a sufficient number of
results from other entities. Thirdly, the current block ID should be the
largest to ensure that only the longest chain is acceptable. This longest
chain principle can effectively avoid forking, resulting in consistent chain
status in a distributed network.

Input: Blk: Received new block;

$\overline{S}$: Local average of received Shapley Value;

$D$: Difficulty in Mining;

Output: ValuationResult: True OR False

1 $S_{t}=Blk.S$; $\overline{S}_{t}=Blk.\overline{S}$;

2 if  _$\|S_{t}-\overline{S}_{t}\|_{p}\leq D$_ then

3 if  _$\|\overline{S}-\overline{S}_{t}\|_{p}\leq D$_ then

4 if  _Blk.ID $\geq$ longest chain length_ then

5 return ValuationResult=ture;

6

7 end if

8 else

9 return ValuationResult=false;

10 end if

11

12 end if

13 else

14 return ValuationResult=false;

15 end if

16

17 end if

18else

19 return ValuationResult=false;

20 end if

Algorithm 2 VerifyBlock (new Blk)

###  4.2 Dynamic Mining Difficulty

The difficulty level in mining new blocks can be adapted dynamically. There
are two main factors influencing the difficulty updates: 1) the total mining
power of the miners and 2) the speed of generating a block. Given the same
mining power, the difficulty level should be decreased as the the block
generation speed increases. Given the same block generation speed, the
difficulty level should be increased as the mining power increases. Difficulty
update can be achieved by deploying a smart contract. For example, in BitCoin,
a block is generated in every ten minutes and the difficulty level is updated
in every two-week duration.

###  4.3 The Payment Scheme

With the FedCoin system in place, an FL model requester starts by depositing
$V$ FedCoins in the FL server. The value of $V$ shall be no greater than the
value of the FL model for the requester. To divide $V$ among FL clients,
blockchain miners, and the FL server, all the entities should register a
transaction account. The value of $V$ is then divided into three parts.

  * •

TrainPrice: payments to the FL clients;

  * •

ComPrice: payment to the FL sever for processing the model aggregation;

  * •

SapPrice: payments to the blockchain network miners for calculating the
Shapley value of each client.

The division can be determined by a pre-agreed smart contract. For example,
the division contract could specify that TrainPrice:ComPrice:SapPrice=7:1:2.
Then, TrainPrice=0.7$V$, ComPrice=0.1$V$, and SapPrice=0.2$V$. The specific
payment scheme is shown in Algorithm 3.

Input: V: The value paid by a model requester;

$\overline{S}$: The final aggragated Shapley Value;

Output: An allocation of $V$

1 while  _FL server receives $V$ from a model requester_ do

2 Calculate TrainPrice and SapPrice;

3 Publish traing task with price TrainPrice;

4 if  _The model is well trained_ then

5 Publish a Shapley task to blockchain network with pirce SapPrice;

6 end if

7

8 end while

9while  _a new block is mined_ do

10 FL server transfers TrainPrice+SapPrice to the block winner; for  _each FL
client $i$_ do

11 if  _$S_{i} >0$_ then

12 $p_{i}=\frac{S_{i}}{\sum_{S_{j}>0}S_{j}}$TrainPrice;

13 block winner transfers $p_{i}$ to $i$;

14 end if

15

16 end for

17

18 end while

Algorithm 3 The Payment Scheme in FedCoin

In Algorithm 3, a model training task is successfully accepted by the FL
server whenever the server receives payment $V$ from the FL model requester.
The payment of $V$ is confirmed when the transfer transaction
(requester$\xrightarrow{V}$server) is recorded in the blockchain. The server
then calculates TrainPrice and SapPrice and leaving ComPrice=V-TrainPrice-
SapPrice as its own payment for processing the task (Line 2). The training
task is then published to FL clients with price TrainPrice (Line 3). When the
training task is completed, the server then publishes a SV calculation task to
the blockchain network with price SapPrice (Line 4-6). As the blockchain
network completes the task by successfully mining a new block, the server
creates a transaction to transfer TrainPrice+SapPrice to the block winner. The
block winner creates the transactions in dividing TrainPrice to the FL clients
with positive Shapley value. All the transactions as well as submitted
unconfirmed transactions are stored in the new block.

##  5 Analysis

Under FedCoin, the decentralized payment scheme is reliable based on the
security of proposed PoSap consensus protocol. Each miner who successfully
calculated the sufficiently converging SV is allowed to record a set of
transactions and receive payment from the FL server. The more mining power
(i.e. resources) a miner applies, the higher its chances to become a block
winner. PoSap provides incentives for miners to contribute their resources to
the system, and is essential to the decentralized nature of the proposed
payment system.

The security of PoSap is also similar to that of the BitCoin system. Empirical
evidence shows that Bitcoin miners may form pools in order to decrease the
variance among their incomes. Within such pools, all members contribute to the
solution of each cryptopuzzle, and share the rewards proportionally to their
contributions. Ideally, a feasible payment system should be designed to resist
the formation of large mining pools. Bitcoin system has been shown to be
vulnerable when a mining pool attracts more than 50% of the miners. Similarly,
the proposed system can also only resist upto 50% of the miners colluding.

Next we discuss how our system fares against the selfish mining strategy Eyal
and Sirer (2018).

###### Observation 1

When the FL server processes FL training model requests sequentially, it is
not rational for colluders to follow the selfish mining strategy.

According to Algorithm 3, each public block winner is paid by the FL server
before creating a new block containing the block reward payment transactions.
When the training task is processed one by one, if a selfish miner becomes the
winner but does not publish this result immedietely, it cannot receive the
block rewards. Meanwhile, the selfish miner cannot mine the next block without
publishing its private block since the next SV task must wait for the
completion of payment in the current block in the setting of sequentially
training models.

###### Observation 2

When the FL server processes FL training model requests in parallel, and all
the miners have the same block propogation delay to the FL server, the
expected revenue for selfish miner is greater than that for honest miners when
the selfish pool attracts more then 25% of the total mining power in the
blockchain network.

If the tasks are published in parallel, a selfish miner can reserve a block
and continue to mine the next block. The state transation and revene analysis
is same as that in Eyal and Sirer (2018), resulting the condition of the
threshold of selfish pool’s mining power to be 1/4 under the condition of the
same propogation delay to the FL server. Thus, processing FL training model
requests in parallel under the proposed scheme is not recommended.

##  6 Experimental Evaluation

To evaluate the proposed FedCoin, we set up a blockchain environment and
design a federated learning task based on a real-world dataset. The objective
of the experiments is to verify whether FedCoin can promote high quality data
from distributed clients and whether the computational cost of PoSap is
feasible.

###  6.1 Experiment Settings

We design our experiment based on a well-known digit classification dataset -
MNIST with 70,000 images and a widely used software environment - TensorFlow -
to perform federated digit classification tasks.

We set up a FL server and 100 FL clients. The MNIST dataset is divided among
the clients such that their data quality vary. We set there are 10 groups of
10 clients each. Clients from each group own datasets which belong to one of
the 10 preset quality levels. We refer to this quality level as a client’s
type. Each client of type $T_{j}$ is randomly assigned a training set
following a uniform distribution over the $10-j$ class labels
($j=0,1,\cdots,9$) as its local dataset. The FL training model in our
experiments is classical neural network. We adopt the popular FedAvg as the FL
aggragtion function, which averages the collected local model parameters to
derive the global FL model parameters. Each local client trains the model for
20 iterations.

Figure 4: The Shapley Values Calculated in FedCoin for Clients in Different
Types

Figure 5: The Shapley Values Calculated in FedCoin for Clients in Different
Types

Figure 6: The Block Generation Time With Mining Varied Difficulty

The consensus nodes are generated based on Docker. Each consensus node can
independantly communicate with each other by sending messages, performing
Shapley value calculation tasks following PoSap, and verifying blocks. The
total computational power is equal to that of our simulation platform (CPU
Interl i7-7700, GPU 2G, RAM 8g, ROM 1t, SSD 256M). We set $p=2$ (Euler
distance) in PoSap for comparing mining difficulty.

###  6.2 Data Quality Evaluation

We adopt Earth Mover’s Distance (EMD) as a metric to measure the data quality
from the perspective of data reliability in this experiment Zhao et al.
(2018). EMD captures the distance for a client’s training data distribution
compared to a given distribution. Here, we use the distribution of the whole
MINIST dataset as the comparison benchmark. A high EMD value for a given
dataset indicates that the datast is of low quality. The data quality measured
by EMD for each client type is shown in Table 2. We can observe that the data
quality linearly decreases from $T_{0}$ to $T_{9}$

Table 2: Data quality of each client type. Data Type | $T_{0}$ | $T_{1}$ | $T_{2}$ | $T_{3}$ | $T_{4}$
---|---|---|---|---|---
Quality (EMD) | 0 | 0.02 | 0.04 | 0.06 | 0.08
Data Type | $T_{5}$ | $T_{6}$ | $T_{7}$ | $T_{8}$ | $T_{9}$
Quality (EMD) | 0.10 | 0.12 | 0.14 | 0.16 | 0.18

###  6.3 Results and Discussion

The average Shapley values calculated by FedCoin for the clients belonging to
each type are shown in Figure 6. The results in Figure 6 show that the
proposed FedCoin approach can accurately compute clients’ Shaply Values which
are critical to incentive mechanisms designed for federated learning. We can
also observe that the computed Shapley values of client type $T_{0}$ is higher
than all the other clients with lower data quality. The Shapley values
decrease with the quality level decays from $T_{1}$ to $T_{9}$. Moreover, only
the values for type $T_{0}$ to $T_{4}$ are positive, indicating that only half
of the clients can positively contribute model accuracy improvement. It also
shows that our PoSap can effectively promote high quality data in
collabratively FL application scenrio. The negative Shapley values for type
$T_{5}$ to $T_{9}$ means the clients can mislead the model training.

To empirically explain why there is negative Shapley values, we train the same
model with clients from each type separately. The testing accuracy of the
model trained by 10 clients in each type is presented in Figure 6. It has been
found that the model accuracy based data from clients with negative Shapley
values is indeed lower than the accuracy achieved in the FL model based on the
100 clients. In our system, we will not reward the clients with negative
Shapley values, discouraging these clients in misleading the model training.

The block generation time with different mining difficulty values is shown in
Figure 6. It is measured in the unit of Shapley calculation iterations, each
of which executing Lines 3 to 14 in Algorithm 1 once. The x-axis contains the
difficulty value in the form of $-log(D)$ (e.g., $x=2$ means $D=1e-2$). The
y-axis contains the corresponding time required to generate a new block
measured in Shapley calculation iterations executed by the block winner. It
can be observed that the block generation time increases as the mining
difficulty increases. As the mining difficulty increases beyond $1e-3$
($D<1e-3$), the block generation time converges at about 2,500 Shapley
calculation iterations. This shows that the computational cost for the
consensus nodes is upper-bounded. In other words, the computational cost of
PoSap is feasible.

##  7 Conclusions

In this paper, we propose FedCoin - a blockchain-based payment system to
enable a federated learning system. It can mobilize free computational
resources in the community to perform costly computing tasks required by FL
incentive schemes. The Shapley value of each FL client, reflecting its
contribution to the global FL model in a fair way, is calculated by the proof
of Shapley (PoSap) consensus protocol. The proposed PoSap which replaces the
traditional hash-based protocol in existing Bitcoin based blockchain systems.
All the payments are recorded in the block in an immutable manner. Under
FedCoin, paying out incentives to FL clients does not need to rely on a
central FL server.

Experimental results show that FedCoin is able to properly determine FL
clients’ Shapley Value-based contribution to the global FL model with an upper
bound on the computational resource required for reaching consensus. To the
best of our knowledge, FedCoin is the first attempt to leverage blockchain
technology in federated learning incentive scheme research. Thereby, it opens
up new opportunities for non-data owners to contribute to the development of
the FL ecosystem Yang et al. (2019b).

## References

  * Ancona et al. [2019] Marco Ancona, Cengiz Öztireli, and Markus H. Gross.  Explaining deep neural networks with a polynomial time algorithm for shapley value approximation.  In Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, pages 272–281, 2019.
  * Bao et al. [2019] Xianglin Bao, Cheng Su, Yan Xiong, Wenchao Huang, and Yifei Hu.  Flchain: A blockchain for auditable federated learning with trust and incentive.  In Proceedings of 5th International Conference on Big Data Computing and Communications, BIGCOM, pages 151–159, 2019.
  * Bonawitz et al. [2017] Keith Bonawitz, Vladimir Ivanov, Ben Kreuter, Antonio Marcedone, H. Brendan McMahan, Sarvar Patel, Daniel Ramage, Aaron Segal, and Karn Seth.  Practical secure aggregation for privacy-preserving machine learning.  In Proceedings of ACM SIGSAC Conference on Computer and Communications Security CCS, pages 1175–1191, 2017.
  * Bonawitz et al. [2019] Keith Bonawitz, Hubert Eichner, Wolfgang Grieskamp, Dzmitry Huba, Alex Ingerman, Vladimir Ivanov, Chloé Kiddon, Jakub Konecný, Stefano Mazzocchi, H. Brendan McMahan, Timon Van Overveldt, David Petrou, Daniel Ramage, and Jason Roselander.  Towards federated learning at scale: System design.  CoRR, abs/1902.01046, 2019.
  * Dillenberger et al. [2019] Donna Dillenberger, Petr Novotny, Qi Zhang, Praveen Jayachandran, Himanshu Gupta, Sameep Mehta, Sandeep Hans, Supriyo Chakraborty, Matthew Walli, John Thomas, et al.  Blockchain analytics and artificial intelligence.  IBM Journal of Research and Development, 2019.
  * Eyal and Sirer [2018] Ittay Eyal and Emin Gün Sirer.  Majority is not enough: bitcoin mining is vulnerable.  Communications of the ACM, 61(7):95–102, 2018.
  * Goodfellow et al. [2016] Ian Goodfellow, Yoshua Bengio, and Aaron Courville.  Deep learning.  MIT press, 2016.
  * Jia et al. [2019] Ruoxi Jia, David Dao, Boxin Wang, Frances Ann Hubis, Nick Hynes, Nezihe Merve Gürel, Bo Li, Ce Zhang, Dawn Song, and Costas J. Spanos.  Towards efficient data valuation based on the shapley value.  In Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics AISTATS 2019, pages 1167–1176, 2019\.
  * Kairouz et al. [2019] Peter Kairouz, H. Brendan McMahan, Brendan Avent, et al.  Advances and open problems in federated learning.  CoRR, abs/1912.04977, 2019.
  * Kang et al. [2019] Jiawen Kang, Zehui Xiong, Dusit Niyato, Shengli Xie, and Junshan Zhang.  Incentive mechanism for reliable federated learning: A joint optimization approach to combining reputation and contract theory.  IEEE Internet of Things Journal, 6(6):10700–10714, 2019.
  * Khan et al. [2019] Latif U. Khan, Nguyen H. Tran, Shashi Raj Pandey, Walid Saad, Zhu Han, Minh N. H. Nguyen, and Choong Seon Hong.  Federated learning for edge networks: Resource optimization and incentive mechanism.  CoRR, abs/1911.05642, 2019.
  * Kim et al. [2019] Hyesung Kim, Jihong Park, Mehdi Bennis, and Seong-Lyun Kim.  Blockchained on-device federated learning.  IEEE Communications Letters, 2019.
  * Li and Cui [2019] Yadong Li and Xin Cui.  Shapley interpretation and activation in neural networks.  CoRR, abs/1909.06143, 2019.
  * McMahan et al. [2017] Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Agüera y Arcas.  Communication-efficient learning of deep networks from decentralized data.  In Proceedings of the 20th International Conference on Artificial Intelligence and Statistics AISTATS, pages 1273–1282, 2017.
  * Nakamoto and others [2008] Satoshi Nakamoto et al.  Bitcoin: A peer-to-peer electronic cash system.  Working Paper, 2008.
  * Ramanan et al. [2019] Paritosh Ramanan, Kiyoshi Nakayama, and Ratnesh Sharma.  Baffle: Blockchain based aggregator free federated learning.  arXiv preprint arXiv:1909.07452, 2019.
  * Song et al. [2019] Tianshu Song, Yongxin Tong, and Shuyue Wei.  Profit allocation for federated learning.  In Proceedings of the 2019 IEEE International Conference on Big Data (IEEE BigData’19), 2019.
  * Yang et al. [2019a] Qiang Yang, Yang Liu, Tianjian Chen, and Yongxin Tong.  Federated machine learning: Concept and applications.  ACM Transactions on Intelligent Systems and Technology (TIST), 10(2):12, 2019.
  * Yang et al. [2019b] Qiang Yang, Yang Liu, Yong Cheng, Yan Kang, Tianjian Chen, and Han Yu.  Federated Learning.  Morgan & Claypool Publishers, 2019.
  * Yu et al. [2020] Han Yu, Zelei Liu, Yang Liu, Tianjian Chen, Mingshu Cong, Dusit Niyato, and Yang Qiang.  A fairness-aware incentive scheme for federated learning.  In Proceedings of the 3rd AAAI/ACM Conference on Artificial Intelligence, Ethics, and Society (AIES-20), 2020.
  * Zhao et al. [2018] Yue Zhao, Meng Li, Liangzhen Lai, Naveen Suda, Damon Civin, and Vikas Chandra.  Federated learning with non-iid data.  CoRR, abs/1806.00582, 2018.


We can introduce a suitable notation for the algebras obtained by such
‘parameterized’ process: all these algebras start from the reals
${\mathbb{R}}$. The first stage adds a new unit $i_{1}$, whose square is equal
to a real parameter $-\eta_{1}$, and we may denote the algebra obtained by a
symbol as ${}_{\eta_{1}}{\mathbb{R}}$, the second stage adds another new unit
$i_{2}$ whose square is equal to another real parameter $-\eta_{2}$, and the
algebra thus obtained may be denoted by a symbol as
${}_{\eta_{2}\eta_{1}}{\mathbb{R}}$, etc. We present the description of the
algebras ${\mathbb{K}}$ obtained through one, two and three stages of the CD
doubling in the following Table, where the appearance of the same algebras as
soon as a single hyperbolic unit is adjoined at some stage is clear:

${\mathbb{K}}$ |  |  Description of ${\mathbb{K}}$ as Cayley-Dickson doubling of ${\mathbb{R}}$ |
---|---|---|---
${\mathbb{R}}$ | Reals | ${\mathbb{R}}$ |
${\mathbb{C}}$ | Complex | ${}_{+\,}{\mathbb{R}}$ |
${\mathbb{C}^{\prime}}$ | Split complex | ${}_{-\,}{\mathbb{R}}$ |
${\mathbb{H}}$ | Quaternions | ${}_{++\,}{\mathbb{R}}$ |
${\mathbb{H}^{\prime}}$ | Split quaternions | ${}_{+-\,}{\mathbb{R}},\ {}_{-+\,}{\mathbb{R}},\ {}_{--\,}{\mathbb{R}}$ |
${\mathbb{O}}$ | Octonions | ${}_{+++\,}{\mathbb{R}}$ |
${\mathbb{O}^{\prime}}$ | Split octonions | ${}_{++-\,}{\mathbb{R}},\ {}_{+-+\,}{\mathbb{R}},\ {}_{-++\,}{\mathbb{R}},\ {}_{+--\,}{\mathbb{R}},\ {}_{-+-\,}{\mathbb{R}},\ {}_{--+\,}{\mathbb{R}},\ {}_{---\,}{\mathbb{R}}$ |

These algebras come in the ordinary CD process endowed with a particular
involutory antiautomorphism, which in each case is the complex, quaternionic
or octonionic conjugation. When the previous family of algebras is endowed
with this antiautomorphism, the $*$-algebras so obtained are enough to
complete the general construction of the Magic Square. But as mentioned
before, the Lie algebras
$({\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}){\mathfrak{su}}_{n}^{l}$ obtained
this way include most but not all real forms of the simple Lie algebras; for
instance those ${\mathfrak{so}}_{2n}^{*}$ with $n$ odd (thus $2n$ is not
multiple of four) do not appear in this family.

Motivated by this, we can ask whether some other slight extension in the CD
doubling will lead to including within a similar scheme all real forms of
simple Lie algebras. The natural freedom still available in the CD process
refers to the construction of an antiinvolution in the extended algebra. The
standard CD process enforces a rigid choice, and while this leads to an
antiinvolution in the extended algebra, this does not give the most general
antiinvolution possible for the algebra obtained at each stage. Of course, if
we adopt another antiinvolution, we will have to accordingly depart from the
properties we have discussed in the standard CD doubling, so neither of these
properties should be taken for granted when we allow further possibilities for
${\mathbb{K}}$; recall also that a $*$ at each stage is required to define the
product in the next stage. Hence our description here is only heuristic.

The next Table describes the possible choices of antiinvolutions which exists
for
${\mathbb{K}}={\mathbb{R}},{\mathbb{C}},{\mathbb{C}^{\prime}},{\mathbb{H}},{\mathbb{H}^{\prime}}$.

${\mathbb{K}}$ |  | Antiinvolution |
---|---|---|---
${\mathbb{R}}$ | Reals | Identity |
${\mathbb{C}}$ | Complex | Complex conjugation |
${\mathbb{C}^{\prime}}$ | Split complex | Complex conjugation |
$\overline{{\mathbb{C}}}$ | Complex | Identity |
$\overline{{\mathbb{C}^{\prime}}}$ | Split complex | Identity |
${\mathbb{H}}$ | Quaternions | Quaternionic conjugation |
${\mathbb{H}^{\prime}}$ | Split quaternions | Quaternionic conjugation |
$\widetilde{\,{\mathbb{H}}\,}\\!$ | Quaternions | Quaternionic reversion |
$\widetilde{{\mathbb{H}^{\prime}}}$ | Split quaternions | Quaternionic elliptic reversion |
$\widehat{{\mathbb{H}^{\prime}}}$ | Split quaternions | Quaternionic hyperbolic reversion |

Here the emphasis should be placed in the fact that the complex or the
quaternionic conjugation are not the unique antiinvolutions in these algebras.
A good notation device is to append an oversymbol characterizing the
involution to the usual symbol name of the algebra. When endowed with ordinary
conjugation, the standard complex or quaternions will be denoted by the usual
letter ${\mathbb{C}},{\mathbb{H}}$, while the remaining possibilities will use
some extra oversymbol.

In particular, for complex numbers (or the split complex) with the identity
antiinvolution the notations
$\overline{{\mathbb{C}}},\overline{{\mathbb{C}^{\prime}}}$ will stand for the
corresponding $*$-algebras; ‘antihermitian’ relative to this antiinvolution
will mean complex antisymmetric. For the quaternions, which are not
commutative, the identity is not an antiinvolution, but further to the
quaternionic conjugation, which changes sign to the three units, there are
another antiinvolutions, called the quaternionic reversion, which changes the
sign to just one of the three units (see [37]). When endowed with this
antiinvolution, quaternions will be denoted $\widetilde{\,{\mathbb{H}}\,}\\!$
and split quaternions give rise to two possibilities, denoted
$\widetilde{{\mathbb{H}^{\prime}}},\widehat{{\mathbb{H}^{\prime}}}$ according
as to which type of unit is reversed.

The new $*$-algebras
$\overline{{\mathbb{C}}},\overline{{\mathbb{C}^{\prime}}},\widetilde{\,{\mathbb{H}}\,}\\!,\widetilde{{\mathbb{H}^{\prime}}},\widehat{{\mathbb{H}^{\prime}}}$
are not nicely normed, and then the sums $z+z^{*}$ will be not real in
general, so for these algebras most properties wich are taken usually for
granted for ${\mathbb{C}},{\mathbb{H}}$ do not hold in the same form. In
particular, as the identity map is not an antiautomorphism for quaternions,
there is no strictly speaking an ‘orthogonal’ quaternionic algebra
${\mathfrak{so}}_{n}{\mathbb{H}}$, and the ‘closest’ quaternionic relative to
the complex
$\overline{{\mathbb{C}}}{\mathfrak{su}}_{n}^{l}\equiv{\mathfrak{so}}_{n}{\mathbb{C}}$
is the algebra $\widetilde{\,{\mathbb{H}}\,}\\!{\mathfrak{su}}_{n}^{l}$, which
precisely gives the expected ‘special unitary interpretation’ for the Lie
algebra ${\mathfrak{so}}_{2n}^{*}$. This interpretation applies in all cases,
whether $n$ is even or odd (notice that when $n$ is even, this algebra had
also another realization over the product of quaternions and split
quaternions).

##  9 From the
$({\mathbb{K}}_{1}\\!\otimes\\!\,{\mathbb{K}}_{2})\mathfrak{sa}_{\kappa_{1}\kappa_{2}\cdots\kappa_{n-1}}(n)$
language to the conventional Lie algebra naming

The Magic Squares can be seen as the ‘direct part’ of a dictionary translating
from the
$({\mathbb{K}}_{1}\,\\!\otimes\\!\,{\mathbb{K}}_{2})\mathfrak{sa}_{\kappa_{1}\kappa_{2}\cdots\kappa_{n-1}}(n)$
language to the standard Lie algebra notation. We end by including the
‘inverse’ dictionary, which lists the real forms of simple Lie algebras and
for each entry identifies their possible
$({\mathbb{K}}_{1}\,\\!\otimes\\!\,{\mathbb{K}}_{2})\mathfrak{sa}_{\kappa_{1}\kappa_{2}\cdots\kappa_{n-1}}(n)$
realizations. As far as we know, this list has not appeared in the previous
literature.

All results will be displayed in a tabular form. We first comment on the
notational conventions which allow to present a large amount of information in
such a compressed form.

First, either ${\mathbb{K}}_{1}$ and ${\mathbb{K}}_{2}$ will be an algebra
taken among
${\mathbb{R}},{\mathbb{C}},{\mathbb{C}^{\prime}},{\mathbb{H}},{\mathbb{H}^{\prime}}$,
endowed with an antiinvolution, which usually (but not always) will be the
ordinary complex, quaternionic conjugation. We take into consideration all
possible choices of a $*$-antiinvolution in the algebras
${\mathbb{R}},{\mathbb{C}},{\mathbb{C}^{\prime}},{\mathbb{H}},{\mathbb{H}^{\prime}}$,
so allowing the different choices for $*$ will correspond to
${\mathbb{R}},{\mathbb{C}},{\mathbb{C}^{\prime}},\overline{{\mathbb{C}}},\overline{{\mathbb{C}^{\prime}}},{\mathbb{H}},{\mathbb{H}^{\prime}},\widetilde{\,{\mathbb{H}}\,}\\!,\widetilde{{\mathbb{H}^{\prime}}},\widehat{{\mathbb{H}^{\prime}}}$,
as listed in the previous section. For these non-octonionic
${\mathbb{K}}_{1},{\mathbb{K}}_{2}$, $n$ can take any value $n\geq 2$.

The numbers ${\kappa_{1},\kappa_{2},\dots,\kappa_{n-1}}$ are a set of real CK
labels, which should be non-zero if the algebra
$({\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}){\mathfrak{su}}_{\kappa_{1}\kappa_{2}\cdots\kappa_{n-1}\\!}(n)$
is to be simple; there is the possibility of getting isomorphic Lie algebras
for different choices of the sequence
${\kappa_{1},\kappa_{2},\dots,\kappa_{n-1}}$. To answer precisely the question
of whether this is the case or not, first consider the $n\choose 2$ members of
the full set of two-index CK labels $\kappa_{ij}$ defined in (4) as
$\kappa_{ij}:=\kappa_{i+1}\kappa_{i+2}\cdots\kappa_{j}$ for $i,j=0,\dots n-1$,
and $i<j$. In the case where all $\kappa_{i}$ are different from zero, all the
$\kappa_{ij}$ are also different from zero. If all $\kappa_{i}$ are positive,
then all the $\kappa_{ij}$ are also positive. But if there are some negative
$\kappa_{i}$, the set of $n\choose 2$ values $\kappa_{ij}$ will include
positive as well as negative values, and the number of $\kappa_{ij}$ which are
negative cannot be arbitrary; an easy checking tells that the number of
negative $\kappa_{ij}$ must necessarily be of the form $l(n-l)$, with $l\leq
n-l$. The number $l$ is an integer in the range between $0$ and $[n/2]$
($[n/2]$ denotes the integer part of $n/2$).

It turns out that two Lie algebras in the family
$({\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}){\mathfrak{su}}_{\kappa_{1}\kappa_{2}\cdots\kappa_{n-1}}(n)$
with different lists of CK labels are only isomorphic when the indices $l$
determined as above are the same (indeed, the number $l$ is precisely the
negative inertia index). In the non-generic cases where at least a
$\kappa_{i}$ vanishes, then some of the two-index CK labels $\kappa_{ij}$
among its $n\choose 2$ members will also vanish, and there are more
possibilities, into which we do not enter here (see a rather complete
discussion in [4, 5, 6]).

We shall use some shorthands to describe the basic lists of labels
$\kappa_{1},\kappa_{2},\dots,\kappa_{n-1}$ associated to any

$({\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}){\mathfrak{su}}_{n}^{l}\equiv({\mathbb{K}}_{1}\\!\otimes\\!{\mathbb{K}}_{2}){\mathfrak{su}}_{\kappa_{1}\kappa_{2}\cdots\kappa_{n-1}}{(n)}.$

Here a list subscripted to ${\mathfrak{su}}$ will always refer to the basic
list of CK labels; when the list is explicitly given $n$ appears within
parentheses and $l$ —which is univocally determined by the list— is not
displayed:

$\\{+\\}$ means that all the $\kappa_{i}$ in the basic list are positive, and
hence reducible to $+1$ by scale changes of the basic coordinates (which
changes each $\kappa_{i}$ by a factor which is necessarily positive). Then the
$n\choose 2$ two-index CK labels $\kappa_{ij}$ are also all positive (and
reducible by scaling to $+1$) and therefore in this case, the inertia index is
equal to zero, $l=0$.

$\\{-\\}$ means that all $\kappa_{i}$ are negative (and hence reducible to
$-1$). In this case there are positive and negative two-index CK labels
$\kappa_{ij}$, and it is clear that there are in all $[n/2]$ negative values
in the list $(1,\kappa_{12},\kappa_{13},\dots\kappa_{1n-1})$ (see the metric
matrix (4)), so that in this case $l=[n/2]$.

$\\{\\!\\!\\{\pm\\}\\!\\!\\}$ refers to the generic case where all
$\kappa_{i}$ are either positive or negative, but no zero values are allowed.
Hence all $\kappa_{i}$ are thus reducible to either $1$ or $-1$.

$\\{\\!\\!\\{n;l\\}\\!\\!\\}$ refers to any particular instance of
$\\{\\!\\!\\{\pm\\}\\!\\!\\}$ where the sequence of $n$ values
$(1,\kappa_{12},\kappa_{13},\dots,\kappa_{1n-1})$ which appear as the diagonal
elements in the metric, has index precisey equal to $l$, this is, there are
precisely $l$ negative terms. An easy checking says that this is equivalent to
saying that the sequence of $\kappa_{i}$ contains no zero and the number of
negative terms in the full set of $\kappa_{ij}$ is $l(n-l)$.

Notice that $\\{+\\}$ or $\\{-\\}$ refer to lists for the $\kappa_{i}$ which
are completely specified, (this is, up to the scaling factors which should
reduce all the $\kappa_{i}$ to $\pm 1$) while $\\{\\!\\!\\{\pm\\}\\!\\!\\}$ or
$\\{\\!\\!\\{n;l\\}\\!\\!\\}$ refer to lists which are only partially
specified, with several concrete possibilities for them. In particular, we
note the following relations:

  * •

$\\{+\\}\equiv{\\{\\!\\!\\{n;0\\}\\!\\!\\}}$.

  * •

$\\{-\\}$ is a member appearing as one of the lists in
${\\{\\!\\!\\{n;[n/2]\\}\\!\\!\\}}$ though for $n>2$ the shorthand
${\\{\\!\\!\\{n;[n/2]\\}\\!\\!\\}}$ refers to several lists, with $\\{-\\}$
being just one of them.

  * •

All the lists in $\\{\\!\\!\\{n;l\\}\\!\\!\\}$ (for any $l$) as well as the
lists $\\{+\\}$ and $\\{-\\}$ are members of $\\{\\!\\!\\{\pm\\}\\!\\!\\}$.

These notational shorthands allow a clear statement on the identification of
$({\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}){\mathfrak{su}}_{\kappa_{1}\kappa_{2}\cdots\kappa_{n-1}}{(n)}$.
All throughout the tables it would be clear that in some cases the Lie algebra
corresponding to a given sequence of the form $\\{\\!\\!\\{\pm\\}\\!\\!\\}$
(for fixed ${\mathbb{K}}_{1},{\mathbb{K}}_{2}$ and $n$) will be (up to
isomorphism) completely independent of the particular set of (non-zero) CK
labels (for instance,
${\mathbb{C}^{\prime}}{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}{(n)}\equiv{\mathfrak{sl}}_{n}{\mathbb{R}}$,
for any choice of non-zero CK labels), while in other cases the Lie algebra
corresponding to a given sequence of the form $\\{\\!\\!\\{\pm\\}\\!\\!\\}$
(for fixed ${\mathbb{K}}_{1},{\mathbb{K}}_{2}$ and $n$) will (up to
isomorphism) depend on the particular set of CK labels through the inertia
index $l$ (as in the orthogonal case
${\mathfrak{so}}_{\\{\\!\\!\\{n;l\\}\\!\\!\\}}{(n)}\equiv{\mathfrak{so}}_{n}^{\cdot\cdot
l\cdot\cdot}$, where the ellipsis in the inertia index tries to remind that
there is a full family of different Lie algebras in the set
${\mathfrak{so}}_{\\{\\!\\!\\{n;l\\}\\!\\!\\}}{(n)}$.

The following five tables contain a very large amount of information, and as
far as we know these tables, which are a kind of reversion (and enlargement)
of the information contained in the Magic Squares, have not appeared before in
the literature.

The disposition tries to be as conventional as possible, but we have to fit
several peculiarities. We first deal with the classical simple Lie algebras,
and the information is splitted in the three Tables I, II and III, each
including the simple Lie algebras in the Cartan series B & D, A and C,
respectively. Next we deal with exceptional simple Lie algebras (all of which
have some octonionic realization as a
$({\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}){\mathfrak{su}}_{\kappa_{1}\kappa_{2}}{(3)}$,
given in Table V, or as a
$({\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}){\mathfrak{su}}{(1)}$), together
with some orthogonal algebras in the Cartan series B & D which allow an
exceptional octonionic realization as
$({\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}){\mathfrak{u}}_{\kappa_{1}}{(2)}$
shown in Table IV.

In the first column, the entries are the simple Lie algebras. For classical
Lie algebras we use the ‘dimension/inertia index’ notation when pertinent,
with the inertia index placed as a superscript. For the exceptional algebras,
we use instead the usual notation ‘rank/Killing Cartan signature’, and the CK
signature is placed also as a superscript but enclosed in parentheses to
minimize the possibility of confusions with the inertia index or any other
misunderstandings.

Only for completeness, we have also included some realizations for which
either ${\mathbb{K}}_{1}$ or ${\mathbb{K}}_{2}$ or both are
${\mathbb{R}},{\mathbb{C}},{\mathbb{C}^{\prime}},{\mathbb{H}},{\mathbb{H}^{\prime}}$
endowed with some antiinvolution which is not the ordinary conjugation, as
explained before.

To keep the tables within reasonable space extent, the
${\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}$ heading of columns, given simply as
${\mathbb{R}},{\mathbb{C}},{\mathbb{H}},{\mathbb{C}}\\!\otimes\\!{\mathbb{C}},{\mathbb{C}}\\!\otimes\\!{\mathbb{H}},{\mathbb{H}}\\!\otimes\\!{\mathbb{H}}$,
cover under each heading the normal and split versions as well as all the
possible antiinvolutions in each algebra. Thus all realizations involving say
${\mathbb{C}}$, ${\mathbb{C}^{\prime}}$, $\overline{{\mathbb{C}}}$ or
$\overline{{\mathbb{C}^{\prime}}}$ as ${\mathbb{K}}_{1}$ (and an invisible
${\mathbb{R}}$ as ${\mathbb{K}}_{2}$) appear under the column heading
${\mathbb{C}}$. The first example in Table I is the entry
$\widetilde{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{n;l\\}\\!\\!\\}}(n)$
appearing in the line of ${\mathfrak{so}}_{2n}^{\cdot\cdot 2l\cdot\cdot}$
under the heading ${\mathbb{H}}$.

A given Lie algebra might have several realizations as
$({\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}){\mathfrak{su}}_{\kappa_{1}\kappa_{2}\cdots\kappa_{n-1}\\!}{(n)}$
and these are placed either in the same line when their
${\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}$ headings are different or spanning
more than one line in the same column when there are several realizations with
the same ${\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}$: the Lie algebra
${\mathfrak{so}}_{4n}^{\cdot\cdot 4l\cdot\cdot}$ is the first appearance of
both instances in the Tables.

As said before, the inertia index appears as a superscript in the usual Lie
algebra name. When a whole family of inertia indices are involved, this is
indicated by enclosing them within an ellipsis; this way
${\mathfrak{so}}_{n}^{\cdot\cdot l\cdot\cdot}$ might refer to either member in
the set of algebras
${\mathfrak{so}}_{n},\,{\mathfrak{so}}_{n}^{1},\,{\mathfrak{so}}_{n}^{2},\dots$;
in some cases to add some specific emphasis, the physicist standard notation
${\mathfrak{so}}(p,q)$ is also indicated, with the usual convention $p>q$; the
relation among both notations is $p+q=n$ and $q=l$. Quite similar notations
apply for the complex unitary algebras ${\mathfrak{su}}$ and the unitary
quaternionic ones ${\mathfrak{sq}}$, which have inertia indices as well as far
as the real forms are concerned.

Again for the completeness sake, we have included here some particular
orthogonal classical Lie algebras which further to their ‘classical
realizations’ as real orthogonal algebras have another octonionic realizations
as well. All these cases appear in the ‘unitary’ (not the ‘special unitary’)
octonionic family which as we mentioned exists only for $n=2$. In this
connection, a last remark to be taken into account is that a given Lie algebra
might appear more than once in the full set of lists. For instance, a
particular algebra like ${\mathfrak{so}}_{16}^{8}$ appears overall in five
lines in the ‘classical part’ shown in Table I and in one line in the
‘octonionic part’ shown in Table IV. In addition to the ‘normal’ entry over
the reals in the first line of the Table I for the classical series B and D,
the ${\mathfrak{so}}_{2n}^{2l}$ algebras with even dimension and even index
have another realization over the reversed quaternions (listed in the second
line), the ${\mathfrak{so}}_{4n}^{4l}$ algebras with dimension and index
multiple of four have still another two realizations over tensor products of
quaternions and reversed quaternions (listed in the third line), and further
to this the split forms ${\mathfrak{so}}_{2n}^{n}$ and
${\mathfrak{so}}_{4n}^{2n}$ have several more realizations (listed in lines
four and five of the ‘classical’ Table I). And furthermore, there is still
another possible realization of ${\mathfrak{so}}_{16}^{8}$ involving
octonions, listed in the last line of the octonionic part covered by Table IV.

Table I.
$({\mathbb{K}}_{1}\\!\otimes\\!{\mathbb{K}}_{2}){\mathfrak{su}}_{\kappa_{1}\cdots\kappa_{n\\!-\\!1}\\!}(n)$
realizations of Lie algebras in classical Cartan series B&D

| $\ \ {\mathbb{R}}$ | ${\mathbb{C}}$ | $\ \ \ \ \ {\mathbb{H}}$ | $\ {\mathbb{C}}\\!\otimes\\!{\mathbb{C}}\\!\\!$ | $\ \ \ \ \ \ {\mathbb{C}}\\!\otimes\\!{\mathbb{H}}$ | $\quad\ \ {\mathbb{H}}\\!\otimes\\!{\mathbb{H}}$
---|---|---|---|---|---|---
${\mathfrak{so}}_{n}^{\cdot\cdot l\cdot\cdot}$ | $\ {\mathbb{R}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{n;l\\}\\!\\!\\}}(n)$ |  |  |  |  |
${\mathfrak{so}}_{2n}^{\cdot\cdot 2l\cdot\cdot}$ | $\ {\mathbb{R}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{2n;2l\\}\\!\\!\\}}(2n)$ |  | $\ \widetilde{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{n;l\\}\\!\\!\\}}(n)\hskip-20.0pt$ |  |  |
${\mathfrak{so}}_{4n}^{\cdot\cdot 4l\cdot\cdot}$ | $\ {\mathbb{R}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{4n;4l\\}\\!\\!\\}}(4n)$ |  | $\ \widetilde{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{2n;2l\\}\\!\\!\\}}(2n)\hskip-20.0pt$ |  |  |  ${\mathbb{H}}\\!\otimes\\!{\mathbb{H}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{n;l\\}\\!\\!\\}}(n)$ $\widetilde{{\mathbb{H}^{\prime}}}\\!\otimes\\!\widetilde{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{n;l\\}\\!\\!\\}}(n)$
${\mathfrak{so}}_{2n}^{n}$ | $\ {\mathbb{R}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{2n;n\\}\\!\\!\\}}(2n)$ |  | $\ \widehat{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)\hskip-20.0pt$ |  |  |
${\mathfrak{so}}_{4n}^{2n}$ | $\ {\mathbb{R}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{4n;2n\\}\\!\\!\\}}(4n)$ |  | $\ \widehat{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2n)\hskip-20.0pt$ |  |  |  ${\mathbb{H}^{\prime}}\\!\otimes\\!{\mathbb{H}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ $\widetilde{\,{\mathbb{H}}\,}\\!\\!\otimes\\!\widetilde{\,{\mathbb{H}}\,}\\!\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ $\widetilde{{\mathbb{H}^{\prime}}}\\!\otimes\\!\widehat{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ $\widehat{{\mathbb{H}^{\prime}}}\\!\otimes\\!\widehat{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$
${\mathfrak{so}}_{2n}^{*}$ |  |  | $\ \widetilde{\,{\mathbb{H}}\,}\\!\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)\hskip-20.0pt$ |  |  |
${\mathfrak{so}}_{4n}^{*}$ |  |  | $\ \widetilde{\,{\mathbb{H}}\,}\\!\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2n)\hskip-20.0pt$ |  |  |  ${\mathbb{H}}\\!\otimes\\!{\mathbb{H}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ $\widetilde{\,{\mathbb{H}}\,}\\!\\!\otimes\\!\widetilde{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ $\widetilde{\,{\mathbb{H}}\,}\\!\\!\otimes\\!\widehat{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$
${\mathfrak{so}}_{n}{\mathbb{C}}$ |  | $\hskip-20.0pt\overline{{\mathbb{C}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)\hskip-20.0pt$ |  |  |  |
${\mathfrak{so}}_{2n}{\mathbb{C}}$ |  | $\hskip-20.0pt\overline{{\mathbb{C}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2n)\hskip-20.0pt$ |  |  |  $\overline{{\mathbb{C}}}\\!\otimes\\!\widetilde{\,{\mathbb{H}}\,}\\!\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ $\overline{{\mathbb{C}}}\\!\otimes\\!\widetilde{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ $\overline{{\mathbb{C}}}\\!\otimes\\!\widehat{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ |

Table II.
$({\mathbb{K}}_{1}\\!\otimes\\!{\mathbb{K}}_{2}){\mathfrak{su}}_{\kappa_{1}\cdots\kappa_{n-1}}(n)$
realizations of classical Lie algebras in Cartan series A

| $\ \ {\mathbb{R}}\ \ \ $ | $\ \ \ {\mathbb{C}}$ | $\ \ {\mathbb{H}}\ \ \ $ | $\ \ \ \ {\mathbb{C}}\\!\otimes\\!{\mathbb{C}}$ | $\ \ \ \ \ {\mathbb{C}}\\!\otimes\\!{\mathbb{H}}$ | $\quad{\mathbb{H}}\\!\otimes\\!{\mathbb{H}}$
---|---|---|---|---|---|---
${\mathfrak{su}}_{n}^{\cdot\cdot l\cdot\cdot}$ |  | ${\mathbb{C}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{n;l\\}\\!\\!\\}}(n)$ |  |  |  |
${\mathfrak{su}}_{2n}^{\cdot\cdot 2l\cdot\cdot}$ |  | ${\mathbb{C}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{2n;2l\\}\\!\\!\\}}(2n)$ |  |  |  ${\mathbb{C}}\\!\otimes\\!{\mathbb{H}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{n;l\\}\\!\\!\\}}(n)$ ${\mathbb{C}}\\!\otimes\\!\widetilde{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{n;l\\}\\!\\!\\}}(n)$ |
${\mathfrak{su}}_{2n}^{n}$ |  | ${\mathbb{C}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{2n;n\\}\\!\\!\\}}(2n)$ |  |  |  ${\mathbb{C}}\\!\otimes\\!{\mathbb{H}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ ${\mathbb{C}}\\!\otimes\\!\widetilde{\,{\mathbb{H}}\,}\\!\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ ${\mathbb{C}}\\!\otimes\\!\widehat{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ |
${\mathfrak{su}}_{2n}^{*}$ |  |  |  |  |  ${\mathbb{C}^{\prime}}\\!\otimes\\!{\mathbb{H}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ ${\mathbb{C}^{\prime}}\\!\otimes\\!\widetilde{\,{\mathbb{H}}\,}\\!\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ |
${\mathfrak{sl}}_{n}{\mathbb{R}}$ |  | ${\mathbb{C}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ |  |  |  |
${\mathfrak{sl}}_{2n}{\mathbb{R}}$ |  | ${\mathbb{C}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2n)$ |  |  |  ${\mathbb{C}^{\prime}}\\!\otimes\\!{\mathbb{H}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ ${\mathbb{C}^{\prime}}\\!\otimes\\!\widetilde{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ ${\mathbb{C}^{\prime}}\\!\otimes\\!\widehat{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ |
${\mathfrak{sl}}_{n}{\mathbb{C}}$ |  |  |  |  ${\mathbb{C}}\\!\otimes\\!{\mathbb{C}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ ${\mathbb{C}}\\!\otimes\\!\overline{{\mathbb{C}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ ${\mathbb{C}^{\prime}}\\!\otimes\\!\overline{{\mathbb{C}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ |  |

Table III.
$({\mathbb{K}}_{1}\\!\otimes\\!{\mathbb{K}}_{2}){\mathfrak{su}}_{\kappa_{1}\kappa_{2}\cdots\kappa_{n-1}}(n)$
realizations of classical Lie algebras in Cartan series C

| $\ \ {\mathbb{R}}\ $ | $\ {\mathbb{C}}\ $ | $\ \qquad{\mathbb{H}}\qquad$ | $\ \\!\\!{\mathbb{C}}\\!\otimes\\!{\mathbb{C}}\\!\\!$ | $\ \ \ \ {\mathbb{C}}\\!\otimes\\!{\mathbb{H}}$ | $\quad{\mathbb{H}}\\!\otimes\\!{\mathbb{H}}$
---|---|---|---|---|---|---
${\mathfrak{sq}}_{n}^{\cdot\cdot l\cdot\cdot}$ |  |  | $\ {\mathbb{H}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{n;l\\}\\!\\!\\}}(n)$ |  |  |
${\mathfrak{sq}}_{2n}^{\cdot\cdot 2l\cdot\cdot}$ |  |  | $\ {\mathbb{H}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{2n;2l\\}\\!\\!\\}}(2n)$ |  |  | $\quad{\mathbb{H}}\\!\otimes\\!\widetilde{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{n;l\\}\\!\\!\\}}(n)$
${\mathfrak{sq}}_{2n}^{n}$ |  |  | $\ {\mathbb{H}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{2n;n\\}\\!\\!\\}}(2n)$ |  |  |  ${\mathbb{H}}\\!\otimes\\!\widehat{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ ${\mathbb{H}^{\prime}}\\!\otimes\\!\widetilde{\,{\mathbb{H}}\,}\\!\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$
${\mathfrak{sp}}_{2n}{\mathbb{R}}$ |  |  | $\ {\mathbb{H}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ |  |  |
${\mathfrak{sp}}_{4n}{\mathbb{R}}$ |  |  | $\ {\mathbb{H}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2n)$ |  |  |  ${\mathbb{H}}\\!\otimes\\!\widetilde{\,{\mathbb{H}}\,}\\!\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ ${\mathbb{H}^{\prime}}\\!\otimes\\!\widetilde{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ ${\mathbb{H}^{\prime}}\\!\otimes\\!\widehat{{\mathbb{H}^{\prime}}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$
${\mathfrak{sp}}_{2n}{\mathbb{C}}$ |  |  |  |  |  $\overline{{\mathbb{C}}}\\!\otimes\\!{\mathbb{H}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ $\overline{{\mathbb{C}}}\\!\otimes\\!{\mathbb{H}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(n)$ |

Table IV.
$({\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}){\mathfrak{su}}_{\kappa_{1}\kappa_{2}}(3)$
octonionic realizations of exceptional Lie algebras in Cartan series E & F

| ${\mathbb{O}}$ | ${\mathbb{C}}\\!\otimes\\!{\mathbb{O}}$ | ${\mathbb{H}}\\!\otimes\\!{\mathbb{O}}$ | ${\mathbb{O}}\\!\otimes\\!{\mathbb{O}}$
---|---|---|---|---
${\mathfrak{f}}_{4}^{(-52)}$ | ${\mathbb{O}}\,{{\mathfrak{su}}}_{\\{+\\}}(3)$ |  |  |
${\mathfrak{f}}_{4}^{(-20)}$ | ${\mathbb{O}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{3;1\\}\\!\\!\\}}(3)$ |  |  |
${\mathfrak{f}}_{4}^{(4)}$ | ${\mathbb{O}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(3)$ |  |  |
${\mathfrak{e}}_{6}^{(-78)}$ |  | $\hskip-10.0pt{\mathbb{C}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{su}}}_{\\{+\\}}(3)$ |  |
${\mathfrak{e}}_{6}^{(-26)}$ |  | $\hskip-10.0pt{\mathbb{C}^{\prime}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(3)$ |  |
${\mathfrak{e}}_{6}^{(-14)}$ |  | $\hskip-10.0pt{\mathbb{C}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{3;1\\}\\!\\!\\}}(3)$ |  |
${\mathfrak{e}}_{6}^{(2)}$ |  | $\hskip-10.0pt{\mathbb{C}}\\!\otimes\\!{\mathbb{O}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(3)$ |  |
${\mathfrak{e}}_{6}^{(6)}$ |  | $\hskip-10.0pt{\mathbb{C}^{\prime}}\\!\otimes\\!{\mathbb{O}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(3)$ |  |
${\mathfrak{e}}_{7}^{(-133)}$ |  |  | ${\mathbb{H}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{su}}}_{\\{+\\}}(3)$ |
${\mathfrak{e}}_{7}^{(-25)}$ |  |  | ${\mathbb{H}^{\prime}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(3)$ |
${\mathfrak{e}}_{7}^{(-5)}$ |  |  |  ${\mathbb{H}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{3;1\\}\\!\\!\\}}(3)$ ${\mathbb{H}}\\!\otimes\\!{\mathbb{O}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(3)$ |
${\mathfrak{e}}_{7}^{(7)}$ |  |  | ${\mathbb{H}}\\!\otimes\\!{\mathbb{O}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(3)$ |
${\mathfrak{e}}_{8}^{(-248)}$ |  |  |  | ${\mathbb{O}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{su}}}_{\\{+\\}}(3)$
${\mathfrak{e}}_{8}^{(-24)}$ |  |  |  | ${\mathbb{O}}\\!\otimes\\!{\mathbb{O}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(3)$
${\mathfrak{e}}_{8}^{(8)}$ |  |  |  |  ${\mathbb{O}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{3;1\\}\\!\\!\\}}(3)$ ${\mathbb{O}^{\prime}}\\!\otimes\\!{\mathbb{O}^{\prime}}\,{{\mathfrak{su}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(3)$

Table V.
$({\mathbb{K}}_{1}\otimes{\mathbb{K}}_{2}){\mathfrak{u}}_{\kappa_{1}}(2)$
exceptional ‘unitary’ octonionic realizations of some particular (orthogonal)
classical Lie algebras in Cartan series B & D and
${\mathbb{O}}{\mathfrak{su}}(1)$ realizations of classical Lie algebras in
Cartan series G

| ${\mathbb{O}}$ | ${\mathbb{C}}\\!\otimes\\!{\mathbb{O}}$ | ${\mathbb{H}}\\!\otimes\\!{\mathbb{O}}$ | ${\mathbb{O}}\\!\otimes\\!{\mathbb{O}}$
---|---|---|---|---
${\mathfrak{so}}_{9}\equiv{\mathfrak{so}}(9)$ | ${\mathbb{O}}\,{{\mathfrak{u}}}_{\\{+\\}}(2)$ |  |  |
${\mathfrak{so}}_{9}^{1}\equiv{\mathfrak{so}}(8,1)$ | ${\mathbb{O}}\,{{\mathfrak{u}}}_{\\{-\\}}(2)$ |  |  |
${\mathfrak{so}}_{9}^{4}\equiv{\mathfrak{so}}(5,4)$ | ${\mathbb{O}^{\prime}}\,{{\mathfrak{u}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2)$ |  |  |
${\mathfrak{so}}_{10}\equiv{\mathfrak{so}}(10)$ |  | $\hskip-10.0pt{\mathbb{C}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{u}}}_{\\{+\\}}(2)$ |  |
${\mathfrak{so}}_{10}^{1}\equiv{\mathfrak{so}}(9,1)$ |  | $\hskip-10.0pt{\mathbb{C}^{\prime}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{u}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2)$ |  |
${\mathfrak{so}}_{10}^{2}\equiv{\mathfrak{so}}(8,2)$ |  | $\hskip-10.0pt{\mathbb{C}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{u}}}_{\\{-\\}}(2)$ |  |
${\mathfrak{so}}_{10}^{4}\equiv{\mathfrak{so}}(6,4)$ |  | $\hskip-10.0pt{\mathbb{C}}\\!\otimes\\!{\mathbb{O}^{\prime}}\,{{\mathfrak{u}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2)$ |  |
${\mathfrak{so}}_{10}^{5}\equiv{\mathfrak{so}}(5,5)$ |  | $\hskip-10.0pt{\mathbb{C}^{\prime}}\\!\otimes\\!{\mathbb{O}^{\prime}}\,{{\mathfrak{u}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2)$ |  |
${\mathfrak{so}}_{12}\equiv{\mathfrak{so}}(12)$ |  |  | $\hskip-10.0pt{\mathbb{H}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{u}}}_{\\{+\\}}(2)$ |
${\mathfrak{so}}_{12}^{2}\equiv{\mathfrak{so}}(10,2)$ |  |  | $\hskip-10.0pt{\mathbb{H}^{\prime}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{u}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2)$ |
${\mathfrak{so}}_{12}^{4}\equiv{\mathfrak{so}}(8,4)$ |  |  |  ${\mathbb{H}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{u}}}_{\\{-\\}}(2)$ ${\mathbb{H}}\\!\otimes\\!{\mathbb{O}^{\prime}}\,{{\mathfrak{u}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2)$ |
${\mathfrak{so}}_{12}^{6}\equiv{\mathfrak{so}}(6,6)$ |  |  | $\hskip-10.0pt{\mathbb{H}^{\prime}}\\!\otimes\\!{\mathbb{O}^{\prime}}\,{{\mathfrak{u}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2)$ |
${\mathfrak{so}}_{16}\equiv{\mathfrak{so}}(16)$ |  |  |  | $\hskip-10.0pt{\mathbb{O}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{u}}}_{\\{+\\}}(2)$
${\mathfrak{so}}_{16}^{4}\equiv{\mathfrak{so}}(12,4)$ |  |  |  | $\hskip-10.0pt{\mathbb{O}}\\!\otimes\\!{\mathbb{O}^{\prime}}\,{{\mathfrak{u}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2)$
${\mathfrak{so}}_{16}^{8}\equiv{\mathfrak{so}}(8,8)$ |  |  |  |  ${\mathbb{O}}\\!\otimes\\!{\mathbb{O}}\,{{\mathfrak{u}}}_{\\{-\\}}(2)$ ${\mathbb{O}^{\prime}}\\!\otimes\\!{\mathbb{O}^{\prime}}\,{{\mathfrak{u}}}_{\\{\\!\\!\\{\pm\\}\\!\\!\\}}(2)$
${\mathfrak{g}}_{2}^{(-14)}$ | ${\mathbb{O}}\,{{\mathfrak{su}}}(1)$ |  |  |
${\mathfrak{g}}_{2}^{(2)}$ | ${\mathbb{O}^{\prime}}\,{{\mathfrak{su}}}(1)$ |  |  |

## Acknowledgments

The initial stages of this work were done in collaboration with F.J. Herranz
[38]. The paper has benefited from several pertinent comments by the referee,
which are gratefully acknowledged.

## Comments on this version

In the present version only very minor changes have been performed over the
the original publication of 2013. In particular, no any change has been made
in the list of references.

## References

  * [1] F. J. Herranz, R. Ortega and M. Santander, Trigonometry of space-times: a new self-dual approach to a curvature/signature (in)dependent trigonometry, J. Phys. A: Math. Gen. 33 (2000), 4525–4551.
  * [2] R. Ortega and M. Santander, Trigonometry of ‘complex Hermitian’ type homogeneous symmetric spaces, J. Phys. A: Math. Gen. 35 (2002), 7877–7917.
  * [3] F. J. Herranz and M. Santander, Homogeneous phase spaces: the Cayley-Klein framework, in Actas de los V Encuentros de Otoño de Geometría Diferencial y Aplicaciones a Física Matemática (Jaca, 1996), eds. J. F. Cariñena, E. Martínez, M. F. Rañada, Memorias de la Real Academia de Ciencias, Madrid, Tomo XXXII (1998), 59–84.
  * [4] J. A. de Azcárraga, F. J. Herranz, J. C. Pérez Bueno and M. Santander, Central extensions of the quasi-orthogonal Lie algebras, J. Phys. A: Math. Gen. 31 (1998), 1373–1394.
  * [5] F. J. Herranz, J. C. Pérez Bueno and M. Santander, Central extensions of the families of quasi-unitary Lie algebras, J. Phys. A: Math. Gen. 31 (1998), 5327–5347.
  * [6] F. J. Herranz, and M. Santander, The family of quaternionic quasi-unitary Lie algebras and their central extensions, J. Phys. A: Math. Gen. 32 (1999), 4495–4507.
  * [7] F. J. Herranz and M. Santander, Casimir invariants for the complete family of quasi-simple orthogonal algebras, J. Phys. A: Math. Gen. 30 (1997), 5411–5426.
  * [8] F. J. Herranz, M. de Montigny, M.A. del Olmo and M. Santander, Cayley-Klein algebras as graded contractions of $so(N+1)$, J. Phys. A: Math. Gen. 27 (1994), 2515–2535.
  * [9] F. J. Herranz and M. Santander, The general solution of the real $Z_{2}^{\otimes N}$ graded contractions of $so(N+1)$, J. Phys. A: Math. Gen. 29 (1996), 6643–6652.
  * [10] A. Ballesteros, F. J. Herranz, M. A. del Olmo and M. Santander, Quantum structure of the motion groups of the two dimensional Cayley-Klein geometries, J. Phys, A: Math. Gen. 26 (1993), 5801–5823.
  * [11] A. Ballesteros, F. J. Herranz, M. A. del Olmo and M. Santander, Quantum (2+1) kinematical algebras: a global approach, J. Phys, A: Math. Gen. 27 (1994), 1283–1297.
  * [12] F. J. Herranz and M. Santander, Cayley-Klein schemes for real Lie algebras and Freudenthal Magic Squares, in GROUP21. Physical Applications and Mathematical Aspects of Geometry, Groups, and Algebras, Volume 1, eds. H.-D. Doebner, W. Scherer and P. Nattermann (World Scientific, Singapore, 1977) pp. 151–155.
  * [13] M. Santander, Symmetric homogeneous spaces in Classical and Quantum Mechanics: a viewpoint for classification, in Proceedings of the First International Workshop on Symmetries in Quantum Mechanics and Quantum Optics, eds. A. Ballesteros, F. J. Herranz, J. Negro, L. M. Nieto and C. M. Pereña (Universidad de Burgos, Burgos, 1999) pp. 335–352.
  * [14] M. Santander, Matrix Models for (exceptional) simple Lie algebras: Two constructions, in Symmetries in Gravity and Field Theory (Homenaje al Profesor José Adolfo de Azcárraga), eds. V. Aldaya, Jose M. Cerveró and Pilar García, (Ediciones Universidad de Salamanca, Salamanca, 2004) pp. 97–114.
  * [15] A. Ballesteros, F. J. Herranz, O. Ragnisco and M. Santander, Contractions, deformations and curvature, Int. J. Theor. Phys. 47 (2008), 649–663.
  * [16] F. J. Herranz and M. Santander, Conformal symmetries of spacetimes, J. Phys. A: Math. Gen. 35 (2002), 6601–6618.
  * [17] F. J. Herranz and M. Santander, Conformal compactification of spacetimes, J. Phys. A: Math. Gen. 35 (2002), 6619–6629.
  * [18] R. Ortega and M. Santander, Trigonometry of the quantum state space, geometric phases and relative phases, J. Phys. A: Math. Gen. 36 (2003), 459–485.
  * [19] R. Gilmore, Lie groups, Lie algebras and some of their applications, (Wiley–Interscience, New York, 1974).
  * [20] E. Cartan, Les groupes réels simples finis et continus, Ann. Sci. Ann. Sci. École Norm. Sup. 31 (1914), 255–262.
  * [21] C. Chevalley, Theory of Lie groups, Princeton Mathematical Series 8, (Princeton University Press, 1946).
  * [22] J. Dieudonné, La geometrie des groupes classiques (Springer, 1970).
  * [23] A. Sudbery, Division algebras, (pseudo)orthogonal groups and spinors, J. Phys. A: Math. Gen. 17 (1984), 939–951,
  * [24] L. E. Dickson, On quaternions and their generalization and the history of the eight square theorem, Ann. Math. 20 (1919), 155–171.
  * [25] M. Postnikov, Lie groups and Lie algebras, Lectures in Geometry. Semester V (Mir, Moscow, 1986).
  * [26] J. Baez, The octonions, Bull. Am. Math. Soc. 39 (2001), 145–205.
  * [27] C. W. Curtis, The four and eight square problem and division algebras, in Studies in modern algebra, ed. A. Albert (Prentice-Hall, Englewood Cliffs, New Jersey, 1963), pp. 100–125.
  * [28] H. Aslaksen, Restricted homogeneous coordinates for the Cayley projective plane, Geom. Dedicata 40 (1991), 245–250.
  * [29] E. Artin, Geometric algebra (Interscience Publishers, New York, 1957).
  * [30] C. Chevalley and R. D. Schafer, The exceptional simple Lie algebras F4 and E6, Proc. Nat. Acad. Sci. USA 36 (1950), 137–141.
  * [31] H. Freudenthal, Lie groups in the foundations of geometry, Adv. Math. 1 (1964), 145–190.
  * [32] M. Santander, Division algebras, composition algebras and triality, in preparation.
  * [33] B. A. Rozenfel’d, The geometry of Lie groups (Kluwer, New York, 1997).
  * [34] A. L. Onishchik and E. B. Vinberg, eds., Lie groups and Lie algebras III (Springer, Berlin, 1991), pp. 167–178.
  * [35] C. Barton and A. Sudbery, Magic squares and matrix models of Lie algebras, Adv. Math. 180 (2003), 596–647.
  * [36] M. Santander and F. J. Herranz, Geometries of orthogonal groups and their contractions: a unified classical deformation viewpoint, Int. J. Mod. Phys. A 12 (1997), 99–107.
  * [37] I. Porteous, Clifford algebras and the classical groups, Cambridge Studies in Advanced Mathematics 50 (Cambridge University Press, Cambridge, 1995).
  * [38] F. J. Herranz, Grupos de Cayley–Klein clásicos y cuánticos, PhD Thesis, Universidad de Valladolid, 1995.