01 Peaceful Days
Though she has not had a good or bad life, Ingeborg has had a peaceful and comfortable life. It is not one of particular happiness or sadness, so it is without a doubt "peaceful".

One could say it is the life any normal person yearns for. However in reality this is not what normal people yearn for; humans want the most, more than anything else. They want to be acknowledged by others, and their deepest wishes and dreams are to be "great". In that way, she lives the life people should want to live, but in reality do not want.

Though this is all a digression, she is content with this life of hers.

02 Catbox
Though she seems like an honest person, she truly is vain at the core. Though her eye was clearly lost long before she ever entered Death City, this means how she lost her eye is an enigma. She accepts this whole heartily, and has a tendency of releasing various conflicting stories to those stupid enough to ask about her eyepatch.

The stories range from "alien abduction" to "Lost in an honorable pistol duel" to various other nonsensical accounts.

Rumor
Someone who constantly invents stories and tales that spread like wildfire across the city or school. They range from minor things to extravagant games that tug at the minds of various people. She will create catboxes that seem supernatural but are in reality the work of a normal human.

For example, the loss of faculty pencils is blamed on the ghost of a long dead teacher, or a invented spirit, etc. This is a tame example, and she tends to do extravagant things during large public events, such as festivals.

Naturally, every tale she crafts is worked so that she can't be implicated.

Eyepatch
Death City is a city not completely ruled by Meisters, rather, it could be said that tourists are a constant stream of revenue and population. To produce more revenue any tourist trap needs merchandise, and Death City is no exception to this pattern. Practically all merchandise is skull themed, naturally, because this is the city of the Shinigami.

However this presents some problems to the local population in the fact that most items are tourist themed. Skulls, skulls everywhere.

To Leif, who is normally laid back, this isn't a problem. However, in the instance of her eyepatch, it is a unique problem. All normal eye patches in the city have skulls on them, which means they'll make whoever wears them look like a pirate. Leif is not a pirate.

As such all her Eye patches are imported and bought from the internet. They aren't novelty items, and are actually pretty expensive. She has taken it as a hobby to own the latest in eye patch fashion.

Shuwen
A name given to her by close friends and some of the faculty in the school. In Japan there was a certain martial arts instructor who said "A Second Strike is Needless", and needless to say he could back up his words.

Though she may not look like it, Leif is a excellent hand to hand combatant, and could easily take the ICMAC in a storm. Her skill and natural aptitude towards Chinese Martial Arts and general hand to hand combat could've given her fame, but because of a simple error during her early fighting days, she fell from grace.

Needless to say, despite being at the level of a normal human in physical attributes, she reflects and enforces the cool discipline of martial arts on her enemies in any sort of combat.

She has actually tried to start a hand to hand combat after school club many times, but it never gets funding or rooms for unknown reasons. If those reasons were found out, she'd probably punch them in the face.

Librarian
Someone who looks after books. It really is as simple as that. In the modern era where computer technology reigns supreme, this job has been sublimed, and even books themselves are fighting a losing battle against electronic reading. However, in Death City she manages the School Library with some measure of pride.

Still, she is a lazy Librarian and tends to throw all her work at various assistants which she hired on using the schools budget. In the end that pride is just vanity, and though she does enjoy books, one of her assistants are probably better at being a Librarian than her.

Good at extorting funding from the higher ups, though. The library has all the latest books, and the best computer lab in the school.

01 Yggdrasil
In Norse myth, the world tree and its roots are collectively called "Yggdrasil". It is the eternal world that holds aloft on its mighty branches the nine worlds. Further, it is the object that survived even the Twilight of the Gods, and humanity survives the end of the world by hiding within it.
In this story, Yggdrasil isn't a tree, it's an organization based somewhere in Scandanavia. It's a legitimate thing created by legitimate means, so one cannot call it "illegal" or a "terrorist organization". It's a recognized faction that holds extraterritoriality in most nations connected to it, and it is accepted as a legitimate institution.
Though the inner workings of the place aren't exactly common knowledge, its ultimate objective is the destruction of [Madness] and [The Sway of Magic], but they constantly try to expand their influence and power.
It is separated into multiple divisions, the most public of which are the Einherjar, Valkyries, and Seior. The first are composed almost entirely of humans with very strong souls; the second those with Grigori Souls, tasked for retrieving those with irregular souls, and the third the Witch corps.

02 The Power of Legacy
The true ability of Omi's family. By turning the memories of each head witch into transmutable data, the abilities of past generations can be passed down seamlessly, only adding on to the new generation. By transferring the memories, and thus the "collective knowledge", a distorted form of immortality is attained. Though one may die, their children will take up the sword and move forward into the future, holding the previous generations fruits.

03 The Sway of Magic
Something that is extremely relevant but is not directly touched upon by Omi. The Sway of Magic and Madness are two related concepts, but in the end they're two different things. While both can be controlled and manipulated, in the end they always end in great destruction and hardship for humanity.
While it may not seem it, The Sway of Magic also dictates and controls the existence of witches by nudging them into destructive results and steady sanity loss. If one cannot honestly dictate their own actions, then they have no choice; and if they have no ability to make their own choices themselves, they're closer to beasts than people. Destruction is self destructive, thus the Sway of Magic will only end horribly for the Witch under its influence and the people around them.
In that way, Omi despises Madness and the Sway of Magic, and considers the Sway of Magic a poison killing the race of witches.

04 Earths No1 Saving System
Secret coalition Yggdrasil is a part of. There are 6 member organizations, one for each area of the world. Omi's is Europe, while the others are located in other areas.
Yggdrasil is the defacto leader, but it really is a loose coalition. They do not keep in regular contact with each other, and the philosophies and morality of the member organizations and their leaders vary greatly. It was created in response to the creation of not only Asura, but also the Great Old Ones as well. It is an anti Madness group in every sense of the word.
Ironically, some of the members and organizations were corrupted or influenced by madness in the past, leading to civil war between members of the coalition.
It is said that among the leaders of the 6, Omi is the strongest. Each leader is a distortion that can't even be called human anymore, and each has a form of Immortality.
Omi's is the immortality of Legacy, and is not conventional immortality.

Numerical Phantasm
Transmigration of numbers. To put it simply, Witches have the capability of shape shifting. That is why their souls are the main ingredient to produce weapons with souls, as their souls allow things to transform. The basis of all magic used by witches can be traced back to this root of "shapeshifting" or "taking materials to make something different". In the case of Omi, she is capable of shape shifting numbers.
To put most of the uses of the ability into simpler terms, the basic idea revolves around percentages, and always making sure the dice is favorable result when it hits the ground. An example would be a bag full of marbles; half of the marbles are white, and the other black. Her ability would be always selecting the black marble if she wanted it.

While it seems like a lackluster ability, in reality is very versatile and useful. However its greatest limitation is that it's unable to make a 1 out of a 0.

Math
An intellectual ability that has nothing to do with magic directly. Her ability to use such a mundane skill is extreme high, and she is capable of creating mathematical equations trivially. Spells or abilities that would normally require Arithmetic magic or something similiar are completely unneeded as she can make the calculations herself.
It really is a wonderfully useful ability that goes with her particular magic style. Very useful for things like spatial and temporal magic.

Soul Envelop
A new revolutionary way of outsmarting the normal and advanced methods of soul detection. Rather than make a barrier, which cuts off the Witches magic and is not foolproof, this ability envelops a soul within a soul.
The basic idea is that instead of using a cardboard cut out of a normal person(barrier) and try to hide behind that, you put a legitimately real person in front of you and hide behind that. As it's a real and legitimate soul, even advanced sensory will be unable to tell that the person under Soul Envelop is a Witch.
Furthermore, as there's no barrier containing or constricting the soul, the Witch is still capable of using magic. It operates on the basis of turning a normal persons soul into data, then unpacking that data outside of the witches soul. The normal person must be willing for this process to work.
It is a revolutionary new step in the warfare between Witches and Meisters, however since Omi is not directly aligned with Witches, they do not have access to this ability.

Prima Materia
A specialized material that is like a black box. It has no special ability or effect on its own, and it is worthless to Witches, but is actually useful as a building block to sorcerers. Currently, Omi is the only one remaining with the proper knowledge to produce it.
As it has no ability of its own other than being a building block for greater things, nothing much can be said about it. However even if it has no special ability it's amazing as a building block and changes for the needs of what it's building seamlessly. Not common knowledge.

Huginn and Muninn
Her wizard robe and hat.
Sentient ravens who assist her by gather information, helping her during combat, and even enhancing her mathematical abilities. Familiars made out of Prima Materia, they are capable of talking but rarely do. They are capable of shifting between the shape of her clothes and that of ravens.

Nameless Spear
Not nameless at all, as Omi calls it Gungnir. It is a spear that is good at being a spear. It has no special effect or ability of its own, but is made out of prima materia.
The one carried on her person is a physical prototype of the real Gungnir that is to be produced. Since it has no special ability other than being troublesome to destroy, it could be said that she has no point in carrying it around.
However, Omi, being her usual vain self, said "I wonder if Gungnir will have use in combat" even if would never actually be used as a weapon in the first place.
The real Gungnir that does not exist yet is capable of "completing" her abilities. If this happens, production of true Aqua Vitae can happen.

Aqua Vitae
The most important thing that Gungnir is a key to unlocking. In alchemy Aqua Vitae is the water of life produced by the philosophers stone. It is said to heal any injury, cure any sickness, and clear the mind of all ailments. Does not exist in this world nor has it existed in this world before.
The best that has been made is a prototype that eases the mind from the Sway of Magic and makes one highly resistant to madness. Omi carries this on her person, but it is not the real thing. Not common knowledge. Its basis is Prima Materia.